H ^ALYTfCAL $ PMCTXCAJU, LLi, * . Ill & 113 WI And sold ty the Princip .^hoiit. tie United States. ' IN MEMORIAM FLOR1AN CAJOR1 - . O o c o t S ?, a A H O o o [Vj w THE IMPROVED SLATED ARITHMETIC. Entered according to Act Of Congress, in the year 1872, by A. S. BARNES & Co., in the Office of the Librarian of Congress, at Washington. SILICATE BOOK SLATE SURFACE. Patented February 24, 1S57 ; January 15, 1867; ami August 25, 1868. JOCELYN'S SLATED BOOK. Patent applied for. BARNES' SLATE AND WATERPROOF FLY-LEAF COMBINATION. Patent applied fur. SCHOOL V ARITHMETIC. ANALYTICAL AND PRACTICAL. BY CHARLES DAVIES, LL.D., [99* DAVIES' PRACTICAL ARITHMETIC, OF THE NEW SERIES, WITH FULL MODERN TRXAT> KENT OF THE SUBJECT, IS OF THE SAME GRADE, AND DESIGNED TO TAKE THK PLACE OF THIS WORK.] A. S. BARNES & COMPANY, NEW YORK, CHICAGO AND NEW ORLEANS, A NEW SERIES OF MATHEMATICS, By CHARLES DAVIES, LL.D., AUTHOR OF THE WEST POINT COURSE OF MATHEMATICS, The following named volumes are entirely new works, written within the past ten years, to conform to all modern improvement, and take the place of the author's older series. NO CONFLICT OP EDITIONS is possible, if patrons will be particular to order the book they want by its exact title. Whenever any change is made so radical as to be likely to cause confusion in classes, THE NAME OF THE BOOK IS CHANGED. Teachers using any work by DAVIES not here-in-after enumerated, are not availing themselves of the advantages offered by THE NEW SERIES. {3^ Primary, Intellectual, and Practical A rithmetics constitute the Series proper. Other volumes are optional. DAVIES' PRIMARY ARITHMETIC. The elementary combinations, by object lessons. DAVIES' INTELLECTUAL ARITHMETIC. Referring all processes to the Unit for analysis. DAVIES' ELEMENTS OF WRITTEN ARITH. Prominently practical, with few rules and explanations. DAVIES' PRACTICAL ARITHMETIC. Complete theory and practice. Substitute for this volume. DAVIES' UNIVERSITY ARITHMETIC. A purely scientific presentation for advanced classes. DAVIES' NE\gKNT|jr ALGEBRA. A connectmgiiBhbetweeBPrithmetic and Algebra. AND A FULL COURSE OF HIGHER MATHEMATICS. Entered according to Act of Congress, in the year 1852, by CHARLES DAVIES, In the Clerk's Office of the District Court of the United States for the Southern District of New York. N. S. A. PREFACE. ARITHMETIC embraces the science of numbers, together with all th rules which are employed in applying the principles of this science to practical purposes. It is -the foundation of the exact and mixed sciences, and the first subject, in a well-arranged course of instruc- tion, to which the reasoning powers of the mind are directed. Because of its great practical uses and applications, it has become the guide and daily companion of the mechanic and man of business. Hence, a full and accurate knowledge of Arithmetic is one of the most im- portant elements of a liberal or practical education. Soon after the publication, in 1848, of the last edition of my School Arithmetic, it occurred to me that the interests of education might be promoted by preparing a full analysis of the science of mathematics, and explaining in connection the most improved methods of teaching. The results of that undertaking were given to the public under the title of "Logic and Utility of Mathematics, with the best methods of in- struction explained and illustrated." The reception of that work by teachers, and by the public generally, is*, strong proof of the deep interest which is felt in any effort, however humble, which may be made to improve our systems of public instruction. In that work a few general principles are laid down to which it is. supposed all the operations in numbers may be referred : 1st. The unit 1 is regarded as the base bfjjfary number, and the consideration of it as the first step in the analysis of every question relating to numbers. 2d. Every number is treated as a collection of units, or as made up of sets of such collections, each collection having its own base, which is either 1, or some number derived from 1. '3d. The numbers expressing the relation between the different units of a number are called the SCALE; and the employment of this term enables us to generalize the laws which regulate the formation of numbers. 4th. By employing the term "fractional units" the same principles are made applicable to fractional numbers ; for, all fractions are but collections of fractional units, these units having a known relation to I. M306011 IV PREFACE. In the preparation of this work, two objects have been kept con- etantly in view: 1st. To make it educational ; and, 2d. To make it practical. To attain these ends, the following plan has been adopted : 1. To introduce every new idea to the mind of the pupil by a sim- ple question, and then to express that idea in general terms under the form of a definition. 2. When a sufficient number of ideas are thus fixed in the mind, they are combined to form the basis of an analysis; so that all the principles are developed by analysis in their proper order. 3. An entire system of Mental Arithmetic has been carried forward with the text, by means of a series of connected questions placed at the bottom of each page; and if these, or their equivalents, are care- fully put by the teacher, the pupil will understand the reasoning in every process, and at the same time cultivate the powers of analysis and abstraction. 4. The work has been divided into sections, each containing a num- ber of connected principles ; and these sections constitute a series of dependent propositions that make up the entire system of principles and rules which the work develops. Great pains have been taken to make the work PRACTICAL in its general character, by explaining^ind illustrating the various applica- tions of Arithmetic in the transactions of business, and by connecting as closely as possible, every principle or rule, with all the applications which belong to it. I have great pleasure in acknowledging my obligations to many teachers who have favored me with valuable suggestions in regard to the definitions, rules, and methods of illustration, in the previous edi- tions. I hope they will find the present work free from the defects they have so kindly pointed out A Key to this volume has been prepared for the use of Teachers onty CONTENTS. JTRST FIVE RULES. Definitions. , 910 Notation and Numeration . . . .' 10 22 Addition of Simple Numbers 2230 Applications in Addition 30 33 Subtraction of Simple Numbers 3337 Applications in Subtraction 37 42 Multiplication of Simple Numbers 42 50 Factors 5053 Applications : 53 56 Division of Simple Numbers 56 61 Equal parts of Numbers 61 64 Long Division 64 68 Proof of Multiplication 6869 Contractions in Multiplication 6971 Contractions in Division 71 74 Applications in the preceeding Rules 74 79 UNITED STATES MONET. United States Money defined w 79 Table of United States Money 79 Numeration of United States Money 80 Reduction of United States Money 8183 Addition of United States Money 8385 Subtraction of United States Money 85 87 Multiplication of United States Money 8791 Division of United States Money 91 93 Applications in the Four Rules 93 96 DENOMINATE NUMBERS. English Money 96 97 Reduction of Denominate Numbers 97 99 Linear Measure 99 101 Cloth Measure 101 102 Land or Square Measure 102104 VI CONTENTS. Cubic Measure or Measure of Volume 104 106 Wine or Liquid Measure *.'. 106108 Ale or Beer Measure 108109 Dry Measure 109110 Avoirdupois Weight 110111 Troy Weight 111112 Apothecaries' Weight 112114 Measure of Time 114116 Circular Measure or Motion 116 Miscellaneous Table 117 Miscellaneous Examples 117 1 19 Addition of Denominate Numbers 1 19 124 Subtraction of Denominate Numbers 124 125 Time between Dates 125 Applications in Addition and Subtraction 126 128 Multiplication .of Denominate Numbers 128 130 Division of Denominate Numbers 130134 Longitude and Time 134 PROPERTIES OF NUMBERS. Composite and Prime Numbers 135 137 Divisibility of Numbers 137 Greatest Common Divisor 137140 Greatest Common Dividend 140142 Cancellation 142145 COMMON FRACTIONS. Definition of, and First Principles 146149 Of the different kinds of Common Fractions 149150 Six Fundamental Propositions < 150 154 Reduction of Common Fractions 154 161 Addition of Common Fractions 161162 Subtraction of Common Fractions 162 164 Multiplication of Common Fractions 164168 Division of Common Fractions : 168172 Reduction of Complex Fractions 172 Denominate Fractions 173176 Addition and Subtraction of Denominate Fractions 176 178 DUODECIMALS. Definitions of, &c 178180 Multiplication of Duodecimals 180182 CONTENTS. VII DECIMAL FRACTIONS. Definition of Decimal Fractions r 182 183 Decimal Numeration First Principles 183 187 Addition of Decimal Fractions 187 191 Subtraction of Decimal Fractions 191193 Multiplication of Decimal Fractions 193 195 Division of Decimal Fractions 195197 Applications in the Four Rules 197 198 Denominate Decimals 198 Reduction of Denominate Decimals 198201 ANALYSIS. General Principles and Methods 201213 RATIO AND PROPORTION. Ratio defined 213214 Proportion , 214216 Simple and Compound Ratio 216218 Single Rule of Three 218223 Double Rule of Three 223228 APPLICATIONS TO BUSINESS. Partnership 228229 Compound Partnership 229231 Percentage 231234 Stock Commission and Brokerage 234237 Profit and Loss 237239 Insurance 239241 Interest 241247 Partial Payments 247251 Compound Interest 251253 Discount 253255 Bank Discount 255257 Equation of Payments 257 260 Assessing Taxes 260 263 Coins and Currency 263 264 Reduction of Currencies 264 265 Exchange , 265268 Duties 268271 Alligation Medial 271272 Alligation Alternate 272276 VIII CONTENTS. INVOLUTION. Definition of, &c '."... 276 EVOLUTION. Definition of, &c 277 Extraction of the Square Root 277 282 Applications in Square Root 282 285 Extraction of the Cube Root 285289 Applications in Cube Root 289 290 ARITHMETICAL PROGRESSION. Definition of, &c. , 290291 Different Cases 291294 GEOMETRICAL PROGRESSION. Definition of, &c 294295 Cases 295297 PROMISCUOUS QUESTIONS. Questions for Practice 298303 MENSURATION. To find the area of a Triangle S03 To find the area of a Square, Rectangle, &c 303 To find the area of a Trapezoid 304 To find the circumference and diameter of a Circle 304 To find the area of a Circle 305 To find the surface of a Sphere 305 To find the contents of a Sphere 305 To find the convex surface of a Prism 306 To find the contents of a Prism 306 To find the convex surface of a Cylinder , 307 To find the contents of a Cylinder To find the contents of a Pyramid To find the contents of a Cone 308 GAUGING. Rules for Gauging 309 APPENDIX. Forms relating to Business in General , 310813 ARITHMETIC DEFINITIONS. 1. A SINGLE THING is called one or a unit. 2. A NUMBER is a unit, or a collection of units. The unit is called the base of the collection. The primary base of every number is the unit one. 3. Each of the words, or terms, one, two, three, four, &c., denotes how many things are taken. These terms are gene- rally called numbers ; though, in fact, they are but the names of numbers. 4. The term, one, has no reference to the kind of thing to which it is applied : and is called an abstract unit. 5. An abstract number is one whose unit is abstract : thus, three, four, six, &c., are abstract numbers. 6. The term, one foot, refers to a single foot, and is called a denominate unit : hence, 7. A denominate number is one whose unit is named, or denominated : thus, three feet, four dollars, five pounds, are denominate numbers. These numbers are also called con- crete numbers. L "What is a single thing called ? 2. What is a number V What is the unit called ? What is the primary base of every number ? a What does each of the words, one, two, three, denote ? What are these words generally called ? W T hat are they, in fact '? 4. Has the term one any reference to the thing to which it may be applied ? What is it called ? 5. What is an abstract number? Give examples of abstract num- bers. 6. What does the term one foot refer to ? What is it called ? 7. What is a denominate number ? Give examples of denominate num- bers. What are denominate numbers, also called ? 10 DEFINITIONS. 8. A SIMPLE NUMBER is a single collection of units. 9. QUANTITY is any thing which can be increased, dimin- ished and measured. 10. SCIENCE treats of the properties and relations of things : ART is the practical application of the principles of Science. 11. ARITHMETIC treats of numbers. It is a science when it makes known the properties and relations of numbers ; and an art, when it applies principles of science to practical pur- poses. 12. A PROPOSITION is something to be done, or demonstrated. 13. An ANALYSIS is an examination of the separate parts of a proposition. 14. An OPERATION is the act of doing something with numbers. The number obtained by an operation is called a result, or answer. 15. A RULE is a direction for performing an operation, and may be deduced either from an analysis or a demonstration. 1C. There are five fundamental processes of Arithmetic : Notation and Numeration, Addition, Subtraction, Multiplica- tion and Division. EXPRESSING NUMBERS. 17. There are three methods of expressing numbers : 1st. By words, or common language ; 2d. By capital letters, called the Roman method ; 3d. By figures, called the Arabic method. 8. What is a simple number ? 9. What is quantity ? 10. Of what does Science treat ? What is Art ? 11. Of what does Arithmetic treat? When is it a science? When an art ? 12. What is a Proposition ? 13. What is an Analysis ? 14. What is an Operation ? What is the number obtained called ? 15. W T hat is a Rule ? How may it be deduced ? 16. How many fundamental rules are there ? What are they ? 17. How many methods are there of expressing numbers? What are they ? NOTATION. 11 BY WORDS. 18. A single thing is called - One. One and one more - Two. Two and one more - Three. Three and one more - Four. Four and one more - .Five. Five and one more - Six. Six and one more - Seven. Seven and one more ' - Eight. Eight and one more - Nine. Nine and one more - Ten. &c. &c. &c. Each of the words, one, two, three, four, Jive, six, &c., denotes how many things are taken in the collection. NOTATION. 19. NOTATION is the method of expressing numbers either by letters or figures. The method by letters, is called Roman Notation; the method by figures is called Arabic Notation. ROMAN NOTATION. 20. In the Roman Notation, seven capital letters are used, viz : I, stands for one ; V, hv five ; X, for ten; L, for fifty ; C, for one hundred ; D, for five hundred', and M, for one thousand. All other numbers are expressed by combining the letters according to the following ROMAN TABLE. I. - - - - One. II. - - - - Two. III. - - - Three. IV. ... Four. V. .-.- Five. VI. ... Six. VII. - - - Seven. VIII. - - - Eight. IX. --- Nine. X. - --- Ten. XX. - - - Twenty. XXX.- - - Thirty. XL. --- Forty. L. - - - Fifty. LX. - - - Sixty. LXX. - . Seventy. LXXX. - - Eighty. XC. - - - Ninety. .---- One hundred. CC. --- Two hundred. CCC. - - - Three hundred. CCCC. - - Four hundred. D. - - - - Five hundred. DC. - - - Six hundred. DCC. - - - Seven hundred. DCCC. - - Eight hundred. DCCCC. . - Nine hundred. M. - - - - One thousand. MD. - - - Fifteen hundred. MM. - - - Two thousand. 12 NOTATION. NOTE. The principles of this Notation are these : 1. Every time a letter is repeated, the number which it denotes is also repeated. 2. If a letter denoting a less number is written on the right of one denoting a greater, their sum will be the number expressed. 3. If a letter denoting a less number is written on the left of one denoting a greater, their difference will be the number ex- pressed. EXAMPLES IN ROMAN NOTATION. Express the following numbers by letters : 1. Eleven. 2. Fifteen. 3. Nineteen. 4. Twenty-nine. 5. Thirty-five. 6. Forty-seven. 7'. Ninety-nine. 8. One hundred and sixty. 9. Four hundred and forty-one, 10. Five hundred and sixty-nine. 11. One thousand one hundred and six, 12. Two thousand and twenty-five. 13. Six hundred and ninety-nine. 14. One thousand nine hundred and twenty-five. 15. Two thousand six hundred and eighty. 16. Four thousand nine hundred and sixty-five. It. Two thousand seven hundred and ninety-one. 18. One thousand nine hundred and sixteen. 19. Two thousand six hundred and forty-one. 20. One thousand three hundred and forty-two. 19. What is Notation ? What is the method by letters called ? What is the method by figures called ? 30. How many letters. are used in the Roman notation? Which are they ? What does each stand for ? NOTE. What takes place when a letter is repeated ? If a letter de- noting a less number be placed on the right of one denoting a greater, how are they read ? If the letter denoting the less number be written on the left, how are they read ? 21. What is Arabic Notation ? How many figures are used? What do they form? Name the figures. How many things does 1 express ? How many things does 2 express ? How many units in 3? In 4 ? In 6 ? In 9 ? In 8 ? What docs express ? What are the other figures called? NOTATION. 13 ARABIC NOTATION. 21. Arabic Notation is the method of expressing numbers by figures. Ten figures are used, and they form the alphabet of the Arabic Notation. They are called zero, cipher, or Naught. 1 One. 2 Two. 3 Three. 4 Four. 5 - Five. 6 - - Six. 7 Seven. 8 - Eight. 9 - - Nine. 1 expresses a single thing, or the unit of a number. 2 two things or two units. 3 three things or three units. 4 four things or four units. 5 five things or five units. 6 six things or six units. 7 seven things or seven units. 8 eight things or eight units. 9 nine things or nine units. The cipher, 0, is used to denote the absence of a thing : Thus, to express that there are no apples in a basket, we write the number of apples is 0. The nine other figures are called significant figures, or Digits. 22. We have no single figure for the number ten. We therefore combine the figures already known. This we do by writing on the right hand of 1, thus : 10, which is read ten. This 10 is equal to ten of the units expressed by 1. It is, however, but a single ten, and may be regarded as a unit, the value of which is ten times as great as the unit 1. It is called a unit of the second order. 22. Have we a separate character for ten ? How do we express ten ? To how many units 1 is ten equal ? May we consider it a single unit ? Of what order ? 14 NOTATION. 23. When two figures are written by the side of each other, the one on the right is in the place of units, and the other in the place of tens, or of units of the second order. Each unit of the second order is equal to ten units of the first order. When units simply are named, units of the first order are always meant. Two tens, or two units of the second order, are written 20 Three tens, or three units of the second order, are written 3Q Four tens, or four units of the second order, are written 40 Five tens, or five units of the second order, are written 50 Six tens, or six units of the second order, are written (50 Seven tens, or seven units of the second order, are written *JQ Eight tens, or eight units of the second order, are written gQ Nine tens, or nine units of the second order, are written 99 These figures are read, twenty, thirty, forty, fifty, sixty, "seventy, eighty, ninety. The intermediate numbers between 10 and 20, between 20 and 30, &c., may be readily expressed by considering their tens and units. For example, the number twelve is made up of one ten and two units. It must therefore be written by setting 1 in the place of tens, and 2 in the place of units : thus, - 12 Eighteen has 1 ten and 8 units, and is written - Jg Twenty-five has 2 tens and 5 units, and is written - - 25 Thirty-seven has 3 tens and 7 units, and is written - 3*7 Fifty-four has 5 tens and 4 units, and is written " - - 54 Hence, any number greater than nine, and less than one hundred, may be expressed by two figures. 24. In order to express ten-units of the second order, or one hundred, we form a new combination. It is done thus, . - 100 by writing two ciphers on the right of 1. This number is read, one hundred. 23. When two figures are written by the side of each other, what is the place on the right called? The place on the left? When units simply are named, what units are meant ? How many units of the second order in 20? In 80? In 40? In 50? In 60? In 70? In 80 ? In 90 ? Of what is the number 12 made up ? Also 18, 25, 37, 54 ? What numbers may be exprsesed by two figures ? NOTATION. 15 Now this one hundred expresses 10 units of the second order, or 100 units of the first order. The one hundred is but an individual hundred, and, in this light, may be regarded as a unit of the third order. We can now express any number less than one thousand. For example, in the number three hundred and . seventy-five, there are 5 units, 7 tens, and 3 hundreds, c g .- Write, therefore, 5 units of the first order, 7 units of the Jj % second order, and 3 of the third * and read from the 375 right, units, tens, hundreds. In the number eight hundred and ninety-nine, there w K - _ are 9 units of the first order, 9 of the second, and 8 of & 3 the third ; ard is read, units, tens, hundreds. ** * o y y In the number four hundred and six, there are 6 units . & of the first order, of the second, and 4 of the third. The right hand figure always expresses units of 4 ' the first order ; the second, units of the second order ; and the third, units of the third order. 25. To express ten units of the third order, or one thous- and, we form a new combination by writing three ciphers on the right of 1 ; thus, 1000 Now, this is but one single thousand, and may be regarded as a unit of the fourth order. Thus, we may form as many orders of units as we please : a single unit of the first order is expressed by 1 , a unit of the second order by 1 and ; thus, 10, a unit of the third order by 1 and two O's ; 100, a unit of the fourth order by 1 and three O's ; 1000, a unit of the fifth order by 1 and four O's ; 10000 ; and so on, for units of higher orders : 24. How do you write one hundred? To how many units of the second order is it equal ? To how many of the lirst order ? May it be considered a single unit ? Of what order is it ? How many units of the third order in 200? In 300? In 400? In 500? In 600? Of what is the number 375 composed ? The number 899 ? The number 406 ? What numbers may be expressed by three figures ? What order of units will each figure express ? 16 NOTATION. 26. Therefore, 1st. The same figure expresses different units according to the place which it occupies : 2d. Units of the first order occupy the place on the right ; units of the second order, the second place ; units of the third order, the third place ; and so on for places still to the left : 3d. Ten units of the first order make one of the second ; ten of the second, one of the third ; ten of the third, one of the fourth ; and so on for the higher orders : 4th. When figures are written by the side of each other, ten units in any one place make one unit of the place next to the left. EXAMPLES IN WRITING THE ORDERS OF UNITS. 1. Write 3 tens. 2. Write 8 units of the second order. 3. Write 9 units of the first order. 4. Write 4 units of the first order, 5 of the second, 6 of the third, and 8 of the fourth. 5. Write 9 units of the fifth order, none of the fourth, 8 of the third, 7 of the second, and 6 of the first. Ans. 90876. 6. Write one unit of the sixth order, 5 of the fifth, 4 of the fourth, 9 of the third, 7 of the second, and of the first. Ans. 7. Write 4 units of the eleventh order. 8. Write forty units of the second order. 9. Write 60 units of the third order, with four of the 2d, and 5 of the first. 10. Write 6 units of the 4th order, with 8 of the 3d, 4 of the 1st. 25. To what are ten units of the third order equal ? How do you write it? How is a single unit of the first order written ? How do you write a unit of the second order ? One of the third ? One of the fourth ? One of the fifth ? 26. On what does the unit of a figure depend ? What is the unit of the first place on the right ? What is the unit of the second place ? What is the unit of the third place ? Of the fourth ? Of the fifth ? Sixth ? How many units of the first order make one of the second ? How many of the second one of the third ? How many of the third one of the fourth, &c. When figures are written by the side of each other, how many units of any place make one unit of the place next to the left? NUMERATION. 17 11. Write 9 units of the 5th order, of the 4th, 8 of the 3d, 1 of the 2d, and 3 of the 1st. 12. Write 7 units of the 6th order, 8 of the 5th, of the 4th, 5 of the 3d, 7 of the 2d, and 1 of the llth. 13. Write 9 units of the 7th order, of the 6th, 2 of the 5th, 3 of the 4th, 9 of the 3d, 2 of the 2d, and 9 of the 1st. 14. Write 8 units of the 8th order, 6 of the 7th, 9 of the 6th, 8 of the 5th, 1 of the 4th, of the 3d, 2 of the 2d, and 8 of the 1st. 15. Write 1 unit of the 9th order, 6 of the 8th, 9 of the 7th, 7 of the 6th, 6 of the 5th, 5 of the 4th, 4 of the 3d, 3 of the 2d, and 2 of the 1st. 16. Write 8 units of the 10th order, of the 9th, of the 8th, of the 7th, 9 of the 6th, 8of the 5th, of the 4th, 3 of the 3d, 2 of the 2d, and of the 1st. 17. Write 7 units of the ninth order, with 6 of the 7th, 9 of the third, 8 of the 2d, and 9 of the 1st. 18. Write 6 units of 8th order, with 9 of the 6th, 4 of the 5th, 2 of the 3d, and 1 of the 1st. 19. Write 14 units of the 12th order, with 9 of the 10th, 6 of the 8th, 7 of the 6th, 6 of the 5th, 5 of the 3d, and 3 of the first. 20. Write 13 units of the 13th order, 8 of the 12th, 7 of the 9th, 6 of the 8th, 9 of the 7th, 7 of the 6th, 3 of the 4th, and 9 of the first. 21. Write 9 units of the 18th order, 7 of the 16th, 4 of the loth, 8 of the 12th, 3 of the llth, 2 of the 10th, 1 of the 9th, of the 8th, 6 of the 7th, 2 of the third, and 1 of the 1st. NUMERATION. 27. NUMERATION is the art of reading correctly any num- ber expressed by figures or letters. The pupil has already been taught to read all numbers from one to one thousand. The Numeration Table will teach him to read any number whatever ; or, to express numbers in words. 27. What is Numeration? What is the unit of the first period? What is the unit of the second ? Of the third ? Of the fourth ? Of the fifth? Sixth? Seventh? Eighth? Give the rale for reading numbers. NUMERATION. NUMERATION TABLE. 6th Period, 5th Period. 4th Period. 3d Period, 2d Period. 1st Period. Quadrillions. Trillions. Billions. Millions. Thousands. Units. II; I ! ! I ! ! I ! ! l-s : ip . ?. * ! ^ 8 -^1 i S3 -25 ||| ||| |a| | , 6, 8 2, 6, 7 5, 879, 023, 301, , . . 123, 087, 7, 000, 735, B . . 4 3, 2 1 0, 460, 548, 000, 087, ( . . 6, 245, 289, 421, 7 2, 549, 1 3 6, 822, 894, 602, 043, 288, 7, 641, 000, 907, 456, 8 4, 912, 876, 4 1 9, 285, 912, 761, 257, 327, 826, 6, 407, 2 1 2, 936, 876, 541, 5 7, 289, 678, 541, 297, 313, 920, 323, 842, 768, 319, 675, NOTES. 1. Numbers expressed by more than three figures are written and read by periods, as shown in the above table. 2. Each period always contains three figures, except the last, which may contain either one, two, or three figures. 3. The unit of the first, or right-hand period, is 1 ; of the second period, 1 thousand ; of the 3d, 1 million ; of the fourth, 1 billion ; and so, for periods, still to the left. 4. To quadrillions succeed quintillions, sextillions, septillions, octillions, &c. 5. The pupil should be required to commit, thoroughly, the names of the periods, so as to repeat them in their regular order from left to right, as well as from right to left. NUMERATION. 19 RULE FOR READING NUMBERS. I. Divide the number into periods of three figures each, beginning at the right hand. II. Name the order of each figure, beginning at the right hand. III. Then, beginning at the left hand, read each period an if it stood alone, naming its unit. EXAMPLES IN READING NUMBERS. 28. Let the pupil point off and read the following numbers -then write them in words. 19. 20. 21. 22. 67 125 6256 4697 23697 412304 7. 8. 9. 10. 11. 12. 6124076 8073405 26940123 9602316 87000032 1987004086 13. 14. 15. 16. 17. 18. 804321049 90067236708 870432697082 1704291672301 3409672103604 49701342641714 8760218760541 904326170365 30267821040291 907620380467026 23. 9080620359704567 24. 9806071234560078 25. 30621890367081263 26. 350673123051672607 NOTE. Let each of the above examples, after being written on the black board, be analyzed as a class exercise ; thus : Ex. 1. How many tens in 67 ? How many units over ? 2. In 125, how many hundreds in the hundreds place? How many tens in the tens place ? How many units in the units place ? How many tens in the number ? 3. In 6256, how many thousands in the thousands place ? How many hundreds in the hundreds place ? How many tens in the tens place ? How many units in the units place ? 4. How many thousands in the number 4697? How many hundreds ? How many tens ? How many units ? 5. How many thousands in the number 23697? How many hundreds ? How many tens ? How many units ? 6. How many hundreds of thousands in 412304? How many ten thousands ? How many thousands ? How many hundreds ? How many tens ? How many units ? 28. Name the units of each order in example 9 ? In 10 ? In 15 ? In 30 ? Give the rule for writing numbers. 20 NUMERATION. RULE FOR WRITING NUMBERS, OR NOTATION. I. Begin at the left hand and write each period in order, as if it icere a period of units. II. When the number of any period, except the left hand period, is expressed by less than three figures, prefix one or two ciphers ; and when a vacant period occurs, fill it with ciphers. EXAMPLES IX NOTATION. 29. Express the following numbers in figures : 1. One hundred arid five. 2.i Three hundred and two. 3. Five hundred and nineteen. _. 4. One thousand and four. 5. Eight thousand seven hundred and one. 6. Forty thousand four hundred and six. / 7. Fifty-eight thousand and sixty-one. 8. Ninety-nine thousand nine hundred and ninety-nine. 9. Four hundred and six thousand and forty-nine. 10. Six hundred and forty-one thousand, seven hundred and twenty-one. 11. One million, four hundred and twenty-one thousands, six hundred and two. 12. Nine millions, six hundred and twenty-one thousands, and sixteen. / ~j 13. Ninety-four millions, eight hundred and seven thous- ands, four hundred and nine. 14. Four billions, three hundred and six thousands, nine hundred and nine. 15. Forty-nine billions, nine hundred and forty-nine thous- ands, and sixty-five. 16. Nine hundred and ninety billions, nine hundred and ninety-nine millions, nine hundred and ninety thousands, nine hundred and ninety-nine. 17. Four hundred and nine billions, two hundred and nine thousands, one hundred and six. 18. Six hundred and forty-five billions, two hundred and sixty-nine millions, eight hundred and fifty-nine thousands, nine hundred and six. NUMERATION. iJl 19. Forty-seven millions, two hundred and four thousands, eight hundred and fifty-one. 20. Six quadrillions, forty-nine trillions, seventy-two bil- lions, four hundred and seven thousands, eight hundred and sixty-one. 21. Eight hundred and ninety-nine quadrillions, four hun- dred and sixty trillions, eight hundred and fifty billions, two hundred millions, five hundred and six thousands, four hun- dred and ninety-nine. 22. Fifty-nine trillions, fifty-nine billions, fifty-nine millions, fifty-nine thousands, nine hundred and fifty-nine. 23. Eleven thousands, eleven hundred and eleven. 24. Nine billions and sixty-five. 25. Write three* hundred and four trillions, one million, three hundred and twentv-one thousands, nine hundred and forty-one. 26. Write nine trillions, six hundred and forty billions, with 7 units of the ninth order, 6 of the seventh order, 8 of the fifth, 2 of the third, 1 of the second, and 3 of the first. 27. Write three hundred and five trillions, one hundred and four billions, one million, with 4 units of the fifth order, 5 of the fourth, 7 of the second, and 4 of the first. 28. Write three hundred and one billions, six millions, four thousands, with 8 units of the fourteenth order. 6 of the third, and two of the second. 29. Write nine hundred and four trillions six hundred and six, with 4 units of the eighteenth order, five of the sixteenth, four of the twelfth, seven of the ninth, and 6 of the fifth. 30. Write sixty-seven quadrillions, six hundred and forty- one billions, eight hundred and four millions, six hundred and forty-four. 31. Write eight hundred and three quintillions, sixty-nine billions, four hundred and forty millions, nine hundred thous- and and three. 32. Write one hundred and fifty-nine sextillions, four hun- dred and five billions, two hundred and one millions, three thousand and six. 33. Write four hundred and four septillions, nine hundred and three sextillions, two hundred and one quintillions, forty quadrillions, and three hundred and four. ADDITION. ADDITION. 30. 1. John has two apples and Charles has three : how many have both ? ANALYSIS. If John's apples be placed with Charles's, there will be five apples. The operation of finding how many apples both have is called Addition. ADDITION TABLE. 2 and are 2 3 and are 3 4 and are 4 5 and are 5 2 and 1 are 3 3 and 1 are 4 4 and 1 are 5 5 and 1 are G 2 and 2 are 4 3 and 2 are 5 4 and 2 are G 5 and 2 are V 2 and 3 are 5 3 and 3 are G 4 and 3 are 7 5 and 3 are 8 2 and 4 are 6 3 and 4 are 7 4 and 4 are 8 5 and 4 are 9 2 and 5 are 7 3 and 5 are 8 4 and 5 are 9 5 and 5 are 10 2 and 6 are 8 3 and 6 are 9 4 and 6 are 10 5 and 6 are 1 1 2 and 7 are 9 3 and 7 are 10 4 and 7 are 11 5 and 7 are 12 2 and 8 are 10 3 and 8 are 11 4 and 8 are 12 5 and 8 are ]3 2 and 9 are 1 1 3 and 9 are 12 4 and 9 are 13 5 and 9 are 14 2 and 10 are 12 3 and 10 are 13 4 and 10 are 14 5 and 10 are 15 6 and are 6 7 and are 7 8 and are 8 9 and are 9 6 and 1 are 7 7 and 1 are 8 8 and 1 are 9 9 and 1 are 10 G and 2 are 8 7 and 2 are 9 8 and 2 are 10 9 and 2 are 11 G and 3 are 9 7 and 3 are 10 8 and 3 are 11 9 and 3 are 12 6 and 4 are 10 7 and 4 are 11 8 and 4 are 12 9 and 4 are 13 G and 5 are 11 7 and 5 are 12 8 and 5 are 13 9 and 5 are 14 6 and 6 are 12 7 and G are 13 8 and 6 are 14 9 and 6 are 15 6 and 7 are 13 7 and 7 are 14 8 and 7 are 15 9 and 7 are 16 G and 8 are 14 7 and 8 are 15 8 and 8 are 16 9 and 8 are 17 6 and 9 are 15 7 and 9 are 16 8 and 9 are 17 9 and 9 are 18 6 and 10 are 16 7 and 10 are 17 8 and 10 are 18 9 and 10 are 19 2. James has 5 marbles and William 7 ? how many have both? 3. Mary has 6 pins and Jane 9 : how many have both ? 4. How many are 4 and 5 and 3 ? 5. How many are 6 and 4 and 9 ? 6. How many are 3 and 7 ? 4 and 6 ? 2 and 8 ? 5 and 5 ? 9 and 1? 10 arid ? and 10? 7. How many are 6 and 3 and 9 ? How many are 18 and 2? 18 and 3? 18 and 5? SIMPLE NUMBERS. 23 8. James had 9 cents and Henry gave him eight more : how many had he in all ? PRINCIPLES AND EXAMPLES. 31. James has 3 apples and John 4 : how many have both ? Seven is called the sum of the numbers 3 and 4. The SUM of two or more numbers is a number which con- tains as many units as all the numbers taken together. ADDITION is the operation of finding the sum of two or more numbers. OF THE SIGNS. 32. The sign + is called plus, which signifies more. When placed between two numbers it denotes that they are to be added together. The sign = is called the sign of equality. When placed between two numbers it denotes that they are equal ; that is, that they contain the same number of units. Thus : 3 + 2 = 5 2+3= how many? 1+2 + 4= how many ? 2 + 3 + 5 + 1= how many? 6 + 7+2+3= how many? 1 + 6 + 7+2 + 3= how many? 1+2+3+4 + 5 + 6 + 7+8 + 9= how many? 1. James has 14 cents, and John gives him 21 : how many will he then have ? OPERATION. 14 ANALYSIS. Having written the numbers, as at the 21 right of the page, draw a line beneath them. oO cents. The first number contains four units and 1 ten, the second 1 unit and two tens. We write the units in one column and the tens in the column of tens. 31. What is the sum of two or more numbers? What is addition ? 32. What is the sign of addition ? What is it called ? What does it signify? Express the sign of equality? When placed between two numbers what does it show ? When is a number equal to the sum of other numbers ? Give an example. 24: ADDITION. We then begin at the right hand, and say 1 and 4 are 5, which we set down below the line in the units' place. We then add the tens, and write the sum in the tens' place. Hence, the sum is 3 tens and 5 units, or 35 cents. OPERATION. 24 2. John has 24 cents, and William 62 : how 62 many have both of them ? gg OPERATION. 3. A farmer has 160 sheep in one field, 20 in 1 ^ another, and 16 in another : how many has he in all ? 196 OPERATION. 4. What is the sum of 328 and 111 ? 499 (5.) (6.) (7.) (8.) 427 329 3034 8094 242 260 6525 1602 330 100 236 103 999 9. What is the sum of 304 and 273 ? 10. What is the sum of 3607 and 4082 ? 11. What is the sum of 30704 arid 471912 ? 12. What is the sum of 398463 and 401536 ? 13. If a top costs 6 cents, a knife 25 cents, a slate 12 cents : what does the whole amount to ? 14. John gave 30 cents for a bunch of quills, 18 cents for an inkstand, 25 cents for a quire of paper : what did the whole cost him ? 15. If 2 cows cost 143 dollars, 5 horses 621 dollars, and 2 yoke of oxen 124 dollars : what will be the cost of them all * 16. Add 5 units, 6 tens, and 7 hundreds. ANALYSTS. We set down the 5 units in the place oi of units, the 6 tens in the place of tens, and the 7 hundreds in the place of hundreds. We then add up, "g ^ JS and find the sum to be 765. We must observe, that in all cases, units of the 5 same order are written in the same column. ^ 6 TT5" SIMPLE NUMBERS. 25 1 7. What is the sum of 3 units, 8 tens, and 4 thousands ? 18. What is the sum of 8 hundreds, 4 tens, 6 units, and 6 thousands ? 19. What is the sum of 3 units, 5 units, 6 tens, 3 tens, 4 hundreds, 3 hundreds, 5 thousands, and 4 thousands? 20. What is the sum of five units of the 4th order, 1 of the 3d, three of the 4th, five of the 3d, and one of the 1st? 21. What is the sum of six units of the 2d order, five of the 3d, six of the 4th, three of the 2d, four of the 3d, two of the 1st, and four of the 2d? 22. What is the sum of 3 and 6, 5 tens and 2 tens, and 3 hundreds and 6 hundreds ? 23. What is the sum of 4 and 5, 5 tens, 3 hundreds and 2 hundreds ? GENERAL METHOD. 33. 1. A farmer paid 898 dollars for one piece of land, and 637 dollars for another; how many dollars did he pay for both ? OPERATION. ANALYSIS. Write the numbers thus, 898 and draw a line beneath them. sum of the units, - 15 sum of the tens, 12 sum of the hundreds, 1 4 sum total 1535 1. The example may be done in another way, thus : Having set down the numbers, as before, OPERATION. say, 7 units and 8 units are 15 units, equal to 898 1 ten and 5 units : set the 5 in the units' place, 63*7 and the 1 ten in the column of tens. Then n say, 1 tea and 3 tens are 4 tens, and 9 tens are 1535 13 tens, equal to 1 hundred and 3 tens. Set the 3 in the tens' place and the 1 hundred in the column of 33. How do you set down the numbers for addition ? Where do you begin to add? If the sum of any column can be expressed by a single figure, what do you do with it? When it cannot, what do you write down ? What do you then add to the next column ? When you add to the next column, what is it called ? What do you set down when you come to the last column ? 26 ADDITION. hundreds. Add the column of hundreds and write down the sum, and the entire sum is 1535. ~ 2. When the sum, in any column, exceeds 9, it produces one or more units of a higher order, which belongs to the next column at the left. In that case, write down the excess over exact tens, and add to the next column as many units of its own order, as there were tens in the sum. This is called carrying to the next column. The number to be carried, should not, in practice, be written under the col- umn at the left, but added mentally. Hence, to find the sum of two or more numbers, we have the following RULE. I. Write the numbers to be added, so that units of the same order shall stand in the same column. II. Add the column of units. Set down the units of the sum and carry the tens to the next column. III. Add the column of tens. Set down the tens of the sum and carry the hundreds to the next column ; and so on, till all the columns are added, and set down the entire sum of the last column. PROOF. 34* The proof of any operation, in Addition, consists In showing that the result or answer contains as many units as there are in all the numbers added, and no more. There are two methods of proof, for beginners.* I. Begin at the top of the units column and add all the columns downwards, carrying from one column to the other, as when the columns were added upwards. If the two results agree the work is supposed to be right. For, it is not likely that the same mistake will have been made in both additions. II. Draw a line under the upper number. Add the lower numbers together, and then add their sum to the upper number. * NOTE. If the teacher prefers the method of proof by casting out the 9's, that method, for the four ground rules, will be found in the University Arithmetic. 84. What does the proof consist of in addition? How many methods of proof are there? Give the two methods. NOTE. Explain the process of addition by reading the figures. SIMPLE NUMBERS. If the last sum is the same as the svm total, first found, the work may be regarded as right. EXAMPLES. 1. What is the sum of the numbers 375, 6321, and 598? The small figure placed under the 4, shows how many are to be carried from the units' column, and the small figure under the 9, how many are to be carried from the tens' column. Also, in the examples below, the small figure un- OPERATION. 375 6321 598 7294 11 der each column shows how many are to be carried to the next column at the left. Beginners should set down the numbers to be carried, as in the examples. Ans. 110012 2221 Ans. (3.) 9841672 793159 888923 11523754 221111 (4.) 81325 6784 2130 Ans. 90239 1110 (5.) 4096 3271 4722 (6.) 9976 8757 8168 9875 9988 8774 (8.) 67954 98765 37214 (9.) 6412 1091 6741 9028 (10.) 90467 10418 91467 41290 (11.) 87032 64108 74981 21360 (12.) 432046 210491 809765 542137 (13.) 21467 80491 67421 4304 2191 (14.) 89479 75416 7647 214 19 (15.) 74167 21094 2947 674 85 (16.) 9947621 704126 81267 9241 495 28 ADDITION. (17.) 34578 ~3750 87 328 17 327 Sum 39087 ~4509 Proof 39087 (20.) 672981043 67126459 39412767 7891234 109126 84172 72120 (18.) 22345 67890 8752 340 350 78 Sum 99755 77410 Proof 1)9755 (21.) 91278976 7654301 876120 723456 31309 4871 978 (19.) 23456 78901 23456 78901 23456 78901 Sum 307071 Proof 307071 (22.) 8416785413 6915123460 31810213 7367985 654321 37853 2685 READING. The pupil should be early taught to omit the intermediate wordi in the addition of columns of figures. Thus, in example 22, instead of saying 5 and 8 are eight and 1 are nine, he should say eight, nine, fourteen, seventeen, twenty. Then, in the column of tens, ten, fifteen, seventeen, twenty-five, twenty-six, thirty-two, thirty-three. This is called reading the columns. Let the pupils be often practised in it, both separately, and in concert in classes. 23. Add 8635, 2194, 7421, 5063, 2196, and 1245 to- gether. 24. Add 246034, 29S765, 47321, 58653, 64218, 5376, 9821, and 340 together. 25. Add 27104, 32547, 10758, 6256, 704321, 730491, 2587316, and 2749104 together. 26. Add 1, 37, 39504, 6890312, 18757421, and 265 to- gether. 27. What is the sum of the following numbers, via: seventy-five; one thousand and ninety-five; six thousand four hundred and thirty-five; two hundred and sixty-seven SIMPLE NUMBERS. 29 thousand ; one thousand four hundred and fifty-five ; twenty- seven millions and eighteen ; two hundred and seventy mil- lions and twenty-seven thousand ? 28. What is the sum of 372856, 404932, 2704793, 9078961, 304165, 207708, 41274, 375, 271, 34, and 6? 29. What is the sum of 4073678, 4084162, 3714567, 27413121, 27049, 87419, 27413, 604, 37, and 9 ? 30. What is the sum of 36704321, 2947603, 999987, 76, 47213694, 21612090, 8746, 31210496, and 3021 ? 31. Add together fifty-eight billions, nine hundred and eighty-two mill ions, four hundred and eighty-seven thousands, six hundred and fifty-four ,- seven hundred and forty billions, three hundred and fifty millions, five hundred and forty thousands, seven hundred and sixty ; four hundred and twenty-five billions, seven hundred and three millions, four hundred and two thousands, six hundred and three ; thirty- four billions, twenty millions, forty thousands and twenty ; five hundred and sixty billions, eight hundred millions, seven hundred thousands and five hundred. (32.) (33.) (34.) 87406 92674 25043 89507 27049 97069 41299 28372 81216 47208 37041 75850 71615 49741 90417 72428 57214 19216 97206 59261 20428 41278 41219 60594 28907 57267 72859 325412 3 40216 43706 S 27049 g 87614 g 21441 28416 92742 87604 72204 87046 71215 70412 90212 . 18972 27426 17618 27042 62081 40261 59876 81697 57274 54301 87489 21859 87415 21642 42673 32018 24672 51814 7268T 30 ADDITION. APPLICATIONS. 35* In all the applications of arithmetic, the numbers ad- ded together must Imve the same unit. In the question, How many head of live stock in a field, there being 6 cows, 2 oxen, 3 steers, and 15 sheep, the unit is 1 head of live stock. And the same principle is applicable to all similar questions. QUESTIONS FOR PRACTICE. 1. HOTT many days are there in the twelve calendar months? January has 31, February 28, March 31, April 30, May 31, June 30, July 31, August 31, September 30, October 31, November 30, and December 31. Ans. 2. What is the total weight of seven casks of merchandise ; No. 1, weighing 960 pounds, No. 2, 725 pounds, No. 3, 830 pounds, No. 4, 798 pounds, No. 5, 698 pounds, No. 6, 569 pounds, No. 7, 987 pounds ? 3. At the Custom House, on the 1st day of June, there ir ere entered 1800 yards of linen; on the 10th, 2500 yards; on the 25th, 600 yards; on the day following, 7500 yards; and the last three days of the month, 1325 yards each day : what was the whole amount entered during the month ? Ans. 4. A farmer has his live-stock distributed in the following manner: in pasture No. 1, there are 5 horses, 14 cows, 8 oxen, and 6 colts ; in pasture No. 2, 3 horses, 4 colts, 6 cows, 20 calves, and 12 head of young cattle; in pasture No. 3, 320 sheep, 16 calves, two colts, and 5 head of young cattle. How much live-stock had he of each kind, and how many Lead had he altogether ? Ans. horses, cows, oxen, colts, calves, head of young cattle, and sheep. Total live-stock, head. 5. What is the interval of time between an event which happened 125 years ago, and one that will happen 267 years hence ? 6. There are 60 seconds in a minute, 3600 in an hour, 35. What principles govern all the additions in Arithmetic ? What is the unit in the question ? How many head of cattle in a pasture ? SIMPLE NUMBERS. 81 86400 in a day, 604800 in a week, 2419200 in a month, and 31557600 in a year: how many seconds in the time named above ? 7. Suppose a merchant to buy the following parcels of cloth: 3912* yards, 1856, 2011, 4540, 937, 6338, 3603, 1586,2044,2951,4228, 1345, 1011,6138,960,607,5150,*, 13886, 617, 7513, 4079, 743, 612, 2519, 1238, and 2445 yards : how many yards in all ? 8 What is the sum of two millions bushels of corn, five hundred and thirty-one thousand bushels, one hundred and twenty bushels, fourteen thousand bushels, thirty thousand and twenty four bushels, five hundred and sixty bushels, and seven hundred and two bushels ? 9 The mail route from Albany to New York is 144 miles, from New York to Philadelphia 90 miles, from Philadelphia to Baltimore 98 miles, and from Baltimore to Washington City 38 miles : what is the distance from Albany to Washing- ton'? 10. A man dying leaves to his only daughter nine hundred and ninety-nine dollars, and to each of three sons two hundred dollars more than he left the daughter. What was each son's portion, and what the amount of the whole estate ? A ( Each son's part dollars. '' \ Whole estate dollars. 11. The number of acres of the public lands sold in 1834 was 4658218 ; in 1835, 12564478 ; in 1836, 25167833 The number sold in 1840 was 2236889; in 1841, 1164796; in 1842, 1 129217 How many acres were sold in the first three, and how many in the last three years ? A C 1st 3 yrs. Ans \ last " 12 What was the population of the British provinces in North America in 1834, the population of Lower Canada being stated at 549005, of Upper Canada 336461, of New ,< Brunswick 152156, of Nova Scotia and Cape Breton 142548, ' of Prince Edward's Island 32292, of Newfoundland 75000 ? Ans. 13. By the census of 1850, the population of the ten largest cities was as follows : New York 515547 ; Philadelphia 340045 ; Baltimore 169054 ; Boston 136881 ; New Orleans 116375; Cincinnati 115436; Brooklyn 96838; St. Louis 32 ADDITION. 77860; Albany 50763; Pittsburgh 46601: what was their entire population ? 14. By the census of 1850, the number of deaf and dumb in the United States was 9803 ; of blind 9794 ; of insane 15610 ; of idiots 15787 : what was the aggregate ? 15. By the census of 1850, the population of the District of Columbia was 51687 ; of the Territory of Minnesota 6077 ; of New Mexico 61547 ; of Oregon 13294 ; of Utah 11380 : what was the population of the Territories, including the District of Columbia ? 16 By the census of 1850, the population of Maine was 583169; of New Hampshire 3L7976; of Vermont 314120; of Massachusetts 994514 ; of Rhode Island 147545 ; and of Connecticut 370792: what was the population of the six New England States ? 17. By the census of 1850, the population of New York was 3097394 ; the population of New Jersey 489555 ; oi Pennsylvania 2311786; and of Delaware 91532 : what was the population of the four Middle States ? 18. By the census of 1 850, the population of Maryland was 583034 ; of Virginia 1421661 ; of North Carolina 869039 ; of South Carolina 668507 ; of Georgia 906185; of Florida 87445; of Alabama 771623; of Mississippi 606526; of Louisiana 517762; and of Texas 212592: what was the whole population of the ten Southern States ? Ans. 19. By the census of 1850, the population of Tennessee was 1002717; of Kentucky 982405; of Ohio 1980329; of Indiana 988416; of Illinois 851470; of Michigan 397654; of Wisconsin 305391 ; of Iowa 192214 ; of Missouri 682044 ; of Arkansas 209897 ; and of California 92597 : what was the entire population of the eleven Western States ? Ans* 20. By the census of 1850, the population of the six New England States was 2728116; of the four Middle States 5990267 ; of the ten Southern States 6644374 ; of the eleven Western States 7685134 ; and of the five Territories 143985 : what was the entire population ? 21. Write the population of each State and Territory, in eluding the District of Columbia, and add the whole as ft single example. SUBTRACTION. SUBTRACTION. 86* 1. John has 3 apples and Charles has 2 : how many have both ? If John's apples be taken from the sum, 5 apples, how many apples will remain ? 2 from 5 leaves how many f 2. If James has 5 apples and gives 3 to Charles, how many will he have left ? 3 from 5 leaves how many $ Let the following table be carefully committed to memory: SUBTRACTION TABLE. 1 from 1 leaves 1 from 2 leaves 1 1 from 3 leaves 2 1 from 4 leaves 3 1 from 5 leaves 4 1 from 6 leaves 5 1 from 7 leaves 6 1 from 8 leaves 7 1 from 9 leaves 8 1 from 10 leaves 9 1 from 11 leaves 10 2 from 2 leaves *0 2 from 3 leaves 1 2 from 4 leaves 2 2 from 5 leaves 3 2 from C leaves 4 2 from 7 leaves 5 2 from 8 leaves 6 2 from 9 leaves 7 2 from 10 leaves 8 2 from 11 leaves 9 2 from 12 leaves 10 3 from 3 leaves 3 from 4 leaves 1 3 from 5 leaves 2 3 from 6 leaves 3 3 from 7 leaves 4 3 from 8 leaves 5 3 from 9 leaves 6 3 from 10 leaves 7 3 from 11 leaves 8 3 from 12 leaves 9 3 from 13 leaves 10 4 from 4 leaves 4 from 5 leaves 1 4 from 6 leaves 2 4 from 7 leaves 3 4 from 8 leaves 4 4 from 9 leaves 5 4 from 10 leaves 6 4 from 11 leaves 7 4 from 12 leaves 8 4 from 13 leaves 9 4 from 14 leaves 10 5 from 5 leaves 5 from C leaves 1 5 from 7 leaves 2 5 from 8 leaves 3 5 from 9 leaves 4 5 from 10 leaves 5 5 from 11 leaves 5 from 12 leaves 7 5 from 13 leaves 8 5 from 14 leaves 9 5 from 15 leaves 10 6 from 6 leaves 6 from 7 leaves 1 6 from 8 leaves 2 6 from 9 leaves 3 6 from 10 leaves 4 6 from 11 leaves 5 6 from 12 leaves 6 from 13 leaves 7 6 from 14 leaves 8 6 from 15 leaves 9 C from 16 leaves 10 7 from 7 leaves 7 from 8 leaves 1 7 from 9 leaves 2 7 from 10 leaves 3 7 from 11 leaves 4 7 from 12 leaves 5 7 from 13 leaves 6 7 from 14 leaves 7 7 from 15 leaves 8 7 from 16 leaves 9 7 from 17 leaves 10 8 from 8 leaves 8 from 9 leaves 1 8 from 10 leaves 2 8 from 11 leaves 3 8 from 12 leaves 4 8 from 13 leaves 5 8 from 14 leaves 6 8 from 15 leaves 7 8 from 16 leaves 8 8 from 17 leaves 9 8 from 18 leaves 10 9 from 9 leaves 9 from 10 leaves 1 9 from 11 leaves 2 9 from 12 leaves 3 9 from 13 leaves 4 9 from 14 leaves 5 9 from 15 leaves 6 9 from 16 leaves 7 9 from 17 leaves 8 9 from 18 leaves 9 9 from 19 leaves 10 34 SUBTRACTION. PRINCIPLES AND EXAMPLES. 37 John has 6 apples and gives 4 to Charles : how many has he left ? The 2 is called the difference between the numbers 6 and 4 and this difference added to the less number 4, will give the greater number 6 : hence, " THE DIFFERENCE between two numbers, is such a number as (added to the less will give the greater. SUBTRACTION is the operation of finding the difference be- tween two numbers. When the two numbers are unequal, the larger is called the minuend, and the less is called the subtrahend. Their difference, whether they are equal or unequal, is called the remainder. OF THE SIGNS. 38 The sign , is called minus, a term signifying less. When placed between two numbers it denotes that the one on the right is to be taken from the one on the left. Thus, 64=2, denotes that 4 is to be taken 'from 6. Here, 6 is the minuend, 4 the subtrahend, and 2 the remainder. 122 = = 12 3= how many ? 16 4= how many ? 11 6= how many ? 18 9= how many? 25 8= how many ? 17 7= how many? 16 8= how many ? 19 9= how many? 20 4= how many ? 137= how many? 14 2= how many? EXAMPLES. 1. James has 27 apples, and gives 14 to John : how many has he left? 37. What is the difference between two numbers ? What is Sub- traction ? What is the larger number called ? What is the smaller number called ? What is the difference called ? In the first exam- ple, which number was the minuend ? Which the subtrahend ? Which the remainder? 38. What is the sign of Subtraction ? What is it called ? What does the term signify ? When placed between two numbers what does it denote ? SIMPLE NUMBERS. 35 The 27 is made up of 7 units and 2 tens; 27 Minuend, and the 14, of 4 units and 1 ten. Subtract 4 ** Q ,. , -. unite from 7 units, and 3 units will remain; 2 subtract 1 ten from 2 tens and 1 ten will re- 13 Bemamder. main : hence, the remainder is 13. 2. What are the remainders in the following examples : (1.) (2.) (3. (4.) Minuends, 874 972 999 8497 Subtrahends, 642 ' 631 367 7487 Remainders, 232 1010 3. A farmer had 378 sheep, and sold 256 : how many had he left? We first write the number 378, and then 256 under 373 it, so that units of the same order shall fall in the same 2 z.a column. We then take 6 units from the 8 units, 5 tens __ from 7 tens, and 2 hundreds from 3 hundreds, leaving for 122 the remainder 122. 4. A merchant had 578 dollars in cash, and paid 475 dol- lars for goods : now much had he left ? 5. What are the remainders in the following examples : (1.) (2.) (3.) 62843 278846 894862 51720 167504 170641 Tll23 39, We see, from the above examples, 1st. That units of the same order are written in the same column ; and 2d. That units of any order are always subtracted from units of the same order. 40. To find the difference when any figure of the minuend is less than the one which stands under it. 1. What is the difference between 843 and 562 ? 39. What principles are shown by the examples ? 40. Can you subtract a greater number from a less? When the tipper figure is the least, how do you proceed? Does this change the difference between the numbers ? What then may we always do ? 36 SUBTK ACTION. ANALYSIS. Begin at the units' column, and say, OPERATION. 2 from 3 leaves 1, which is written in the units' g^o place. At the next place we meet a difficulty, for we cannot subtract a greater number from a less. If now, we take 1 from the 8 hundreds (equal to f 10 tens) and add it to the 4 tens, the minuend will become 7 hundreds, 14 tens, and 3 units, as written below. We may then say 6 tens from 14 tens leaves 8 tens ; and then 5 hundreds from 7 hundreds leaves 2 hundreds ; hence, the remainder is 281. The same result is obtained by adding, mentally, 10 to 1 o the 4 tens, and then adding 1 to 5, the next figure of the subtrahend at the left ; for, adding 1 to the 5 is the same 562 as diminishing the 8 by 1. This process of adding 10 _J to a figure of the minuend and returning 1 to the next 281 figure of the subtrahend, at the left, is called 'borrowing. 41* Hence, to find the difference between two numbers, we have the following KULE. I. Set down the less number under the greater, so that units of the same order shall fall in the same column. IL Begin at the right hand subtract each figure of the lower line from the one directly over it, when the upper figure is the greater; but when it is the less, add 10 to it, before subtracting, after which add 1 to the next figure of the subtrahend. PROOF. The remainder or difference is such a number as added to the subtrahend, will give a sum equal to the minuend, (Art. 7,) hence : Add the remainder to the subtrahend. If the work is right $IK sum will be equal to the minuend. EXAMPLES. d^ Minuends, Subtrahends, Remainders, Proofs, (1.) 8592678 1078953 J 2 -> 67942139 9756783 (3.) 219067803 104202196 7513725 8592678 67942139 219067803 41. How do you set down the numbers for subtraction ? Where do you begin to subtract ? How do you subtract ? Give the rule ? How do you prove subtraction? SIMPLE NUMBERS. 37 (4.) (5.) (6.) (7.) (8.) 10000 30000 67087 100000 87000 4 9999 40000 1 1009 Remainders, 9996 85991 9. From 2637804 take 2376982. 10. From 3762162 take 826541. 11. From 78213609 take. 27821890. 12. From thirty thousand and ninety-seven, take one thousand six hundred and fifty-four. 13. From one hundred millions two hundred and forty-seven thousand, take one million four hundred and nine. 14. Subtract one from one million. 15. From 804367 subtract 27905. 16. From 18623041 subtract 61294. 17. From 4270492 subtract 26409. 18. From 8741209 subtract 728104. 19. From 741874 subtract 689346. SPELLING READING. 42. 1. What is the difference between 725 and 341 ? OPERATION. By the common method, which is spelling, we say, 725 1 from 5 leaves 4 ; 4 from 12 leaves 8 ; 1 to carry 34^ to 3 is 4 ; 4 from 7 leaves 3. Reading the words which express the final result, we should make the operations mentally, and say, 4, 8, 3. Let the pupils be practiced separately in the reading, and also in concert in classes. APPLICATIONS. 43. It should be observed, that in all the applications of Subtraction, one number can be subtracted from another, only when they both have the same unit. . 42. Explain the process of reading the results in subtraction. 43. What is always necessary in order that one number may be subtracted from another ? 38 SUBTEACTIOK. EXAMPLES FOR PRACTICE. 1. Suppose John were Lorn in eighteen hundred and fifteen, and James in eighteen hundred and twenty-five : what is the difference of their ages ? 2. A man was born in 1785 : what was his age in 1830 ? Ans. 3. Suppose I lend a man 1565 dollars, and he dies, owing me 450 dollars : how much had he paid me? Ans, 4. In five bags are different sums of money to the amount in all of 1000 dollars. In the first there are 100 dollars; in the second, 314 dollars; in the third, 143 dollars ; and in the fourth, 209 dollars : how many dollars does the fifth contain ? Ans. 5. America was discovered by Christopher Columbus in the year 1492. What number of years has since elapsed ? 6. George Washington was born in the year 1732, and died in 1799 : how old was he at the time of his death ? Ans. 7. The declaration of independence was published, July 4th, 1776: how many years to July 4th, 1838? Ans. 8. In 1850 there were in the State of New York 3,097,394 inhabitants, and in the State of Pennsylvania 2,311,786 in- habitants: how many more inhabitants were there in New York than in Pennsylvania ? Ans. 9. The revolutionary war began in 1775 ; the next war in 1812 : what time elapsed between their commencements? Ans. 10. In 1850 there were in New York, which is the largest city in the United States, 515,547 inhabitants, and in Phila- delphia, the next largest city, 340,045: how many more inhabitants were there in New York than in Philadelphia ? Ans. 11. A man dies worth 1200 dollars: he leaves 504 to his daughter, and the remainder to his son? what was the son's portion ? 12. Suppose a gentleman has an income of 3090 dollars a year, and pays for taxes 150 dollars, and expends besides 307 dollars: how much does he save? SIMPLE NUMBERS. 39 IS. A merchant bought 500 barrels of flour for 3500 dol- lars; he sold 250 barrels for 2000 dollars: how many bar- rels remained on hand, and how much must he sell them for, that he may lose nothing ? 14. The tune of Yankee Doodle was composed by a doctor of the British Army to ridicule the Americans in 1775 : how many years to the present time ? 15. Lord Corn wallis surrendered at Yorktown, and marched into the American lines in 1781 to the tune of Yankee Doodle: how many years was that after the tune was composed? Am. 16. At a certain period there were 4338472 children in the United States between the ages of 5 and ]5; of this number 2477667 were in schools: how many were out of schools? 17. The circulation of the blood was discovered in 1616: how many years to 1855? 18. Henry Hudson sailed up the Hudson river in 1609: how many yean, since? 19. Pliny the historian died 17 years after the birth of Christ: how many years before the declaration of independ- ence ? Ans. 20. Potatoes were carried to Ireland from America in 1565 : how many years was that before the settlement of Plymouth in 1620? 21. The Mariner's Compass was discovered in England in the year 1302 : how many years was this before the discovery of America in 1492 ? How many years to the present time? Ans. 22. A merchant bought 1675 yards of cloth, for which he paid 5025 dollars: he then sold 335 yards for 1005 dollars; how much had he left, and what did it cost him ? Ans. 23. In 1850 the slaves in the United States amounted to 3204313; free colored to 434495: what was their differ- ence? 24. What length of time elapsed between the birth of William Penn in 1644 and the birth of Sir William Herschel in 1738? 40 SUBTRACTION. 25. What length of time elapsed between the birth of Sir Francis Bacon in 1561 and the birth of Benjamin Franklin in 1706? 26. What length of time elapsed between the birth of Shakespeare in 1564 and the birth of George Washington in 1732? 27. What length of time elapsed between the birth of John Milton in 1608 and the Declaration of Independence in 1776? 28. What length of time elapsed between the birth of Oliver Cromwell in 1599 and the birth of Patrick Henry in 1736? 29. By the census of 1850, the number of white inhabitants in the United States amounted to 19553068 ; and the blacks to 3638808 : by how many did the white inhabitants exceed the black ? 30. By the census of 1850, the entire population of the United States was 23191876; that of the six New England States, 2728116: by how many did the whole population exceed that of the six New England States ? 31. In 1850, the slaves in the United States amounted to 3204313; and the free colored to 434495: what was their difference ? APPLICATIONS IN ADDITION AND SUBTRACTION. 1. A merchant buys 19576 yards of cloth of one person, 27580 yards of another, and 375 of a third ; he sells 1050 yards to one customer, 6974 yards to another, and 10462 yards to a third : how many yards has he remaining ? Ans. 2. A person borrowed of his neighbor at one time 355 dollars, at another time 637 dollars, and 403 dollars at another time; he then paid him 977 dollars; how much did he owe him? 3. I have a fortune of 2543 dollars to divide amoncj my four sons, James, John, Henry and Charles. I give James 504 dollars, John 600 dollars, and Henry 725 : how much remains for Charles? 4. I have a yearly income of ten thousand dollars. I pay 275 for rent, 220 dollars for fuel, 35 dollars to the doctor, and 3675 dollars for all my other expenses: how much have I left at the end of the t year ? SIMPLE NUMBERS. 41 5. A man pays 300 dollars for 100 sheep, 95 dollars for a pair of oxen, 60 dollars for a horse, and 125 dollars for a chaise. He gives 100 bushels of wheat worth 125 dollars, a cow worth 25 dollars, a colt worth 40 dollars, and pays the rest in cash : how much money does he pay ? 6. A merchant owes 450120 dollars, and has property as follows : bank stock 350000 dollars, western lands valued at 225100, furniture worth 4000 dollars, and a store of goods worth 96000: how much is he worth? Ans. 7. If a man's income is 3467 dollars a year, and he spends 269 dollars for clothing, 467 for house rent, 879 for provi- sion, and 146 for travelling: how much will he have left at the end of the year? 8. A man gains 367 dollars, then loses 423 ; a second time he gains 875 and loses 912 ; he then gains 1012 dollars ; how much more has he gained than lost? 9. If I agree to pay a man 36 dollars for plowing 25 acres of land, 200 dollars for fencing it, and 150 for cultivating it, how much shall I owe him after paying 331 dollars ? Ans. 10. A merchant bought 85 hogsheads of sugar for 28675 dollars, paid 1231 dollars freight, and then sold it for 1683 dollars less than it cost him : how much did he receive for it? 11. If I buy 489 oranges for 912 cents, and sell 125 for 186. cents, and then sell 134 for 199 cents, how many will be left, and how much will they have cost me ? 12. By the census of 1850, the entire population of the United States was 23191876 ; the slave population 3204313 ; free colored 434495 : what was the white population ? Ans. 13. Six men bought a tract of land for 36420 dollars: the first man paid 12140 ; the second 3035 less than the first; the third 346 ; the fourth 6070 more than the third ; the fifth 1821 less than the fourth : how much did the sixth man pay ? 14. The coinage in the United States Mint from its establishment in the year 1792 to 1836 was thus: gold 22102035 dollars; silver 46739182 dollars; copper 740331 dollars. The amount coined from the year 1837 to 1848 was 81436165 dollars: how much more'was coined in the last mentioned period than in the first? MULTIPLICATION. MULTIPLICATION. 44. 1. If Charles gives 2 cents apiece for two oranges, how much do they cost him ? 2. If Charles gives 2 cents apiece for three oranges, how much do they cost him ? 3. If he gives 2 cents apiece for 4 oranges, how much do they cost him ? The cost, in each case, may be obtained by adding the price of a single orange : .2 + 2 = 4 cents, the cost of 2 oranges. 2+2+2=6 cents, the cost of 3 oranges. 2 + 2 + 2 + 2 = 8 cents, the cost of 4 oranges. In toe first case 2 is taken two times ; in the second, three times; in the third, four times; and any number may be repeated by adding it continually to itself. MULTIPLICATION TABLE. Once is 3 times are 5 times are Once 1 is 1 3 times 1 are 3 5 times 1 are 5 Once 2 is 2 3 times 2 are 6 5 times 2 are 10 Once 3 is 3 3 times 3 are 9 5 times 3 are 15 Once 4 is 4 3 times 4 are 12 5 times 4 are 20 Once 5 is 5 3 times 5 are 15 5 times 5 are 25 Once 6 is G 3 times 6 are 18 5 times 6 are 30 Once 7 is 7 3 times 7 are 21 5 times 7 are 35 Once 8 is 8 3 times 8 are 24 6 times 8 are 40. Once 9 is 9 3 times 9 are 27 5 times 9 are 45 Once 10 is 10 3 times 10 are 30 5 times 10 are 50 Once 11 is 11 3 times 11 are 33 5 times 1 1 are 55 Once 12 is 12 3 times 12 are 36 5 times 12 are 60 2 times are 4 times are 6 times are 2 times 1 are 2 4 times 1 are 4 6 times 1 are 6 2 times 2 are 4 4 times 2 are 8 6 times 2 are 12 2 times 3 are 6 4 times 3 are 12 6 times 3 are 18 2 times 4 are 8 4 times 4 are 16 6 times 4 are 24 2 times 5 are 10 4 times 5 are 20 6 times 5 are 30 2 times 6 are 12 4 times 6 are 24 6 times 6 are 36 2 times 7 are 14 4 times 7 are 28 6 times 7 are 42 2 times 8 are 16 4 times 8 are 32 6 times 8 are 48 2 times 9 are 18 4 times 9 are 36 6 times 9 are 54 2 times 10 are 20 4 times 10 are 40 6 times 10 are 60 2 times 11 are 22 4 times 11 are 44 6 times 11 are 66 2 times 12 are 24 4 times 12 are 48 6 times 12 are 72 SIMPLE NUMBERS. 7 times are 9 times are 11 times are 7 times 1 are 7 9 times 1 are 9 11 times 1 are 11 7 times 2 are 14 9 times 2 are 18 11 times 2 are 22 7 times 3 are 21 9 times 3 are 27 11 times 3 are 33 7 times 4 are 28 9 times 4 are 36 11 times 4 are 44 7 times 5 are 85 9 times 5 are 45 11 times 5 are 55 7 times 6 are 42 1 9 times G are 54 11 times 6 are 66 7 times 7 are 49 } 9 times 7 are 68 11 times 7 are 77 7 times 8 are 56 9 times 8 are 72 11 times 8 are 88 7 times 9 are 63 9 times 9 are 81 11 times 9 are 99 7 times 10 are 70 9 times 10 are 90 11 times 10 are 110 7 times 11 are 77 9 times 11 are 99 11 timfes 11 are 121 7 times 12 are 84 9 times 12 are 108 11 times 12 are 132 8 times are 10 times are 12 times are 8 times 1 are 8 10 times 1 are 10 12 times 1 are 12 8 times 2 are 16 10 times 2 are 20 12 times 2 are 24 8 times 3 are 24 10 times 3 are 30 12 times 3 are 36 8 times 4 are 32 ^10 times 4 are 40 12 times 4 are 48 8 times 5 are 40 "lO times 5 are 50 12 times 5 are 60 8 times 6 are 48 10 times 6 are 60 12 times 6 are 72 8 times 7 are 56 10 times 7 are 70 12 times 7 are 84 8 times 8 are 64 10 times 8 are 80 12 times 8 are 96 8 times 9 are 72 10 times 9 are 90 -12 times 9 are 108 8 times 10 are 80 10 times 10 are 100 12 times 10 are 12G 8 times 11 are 88 10 times 11 are 110 12 times 11 are 132 j 8 times 12 are 96 10 times 12 are 120 12 times 12 are 144 4. What is the cost of 6 yards of ribbon at 7 cents a yard ? ANALYSIS. Six yards of ribbon will cost 6 times as much as 1 yard. Since 1 yard costs 7 cents, 6 yards will cost 6 times 7 cents, which are 42 cents. Let the pupil analyze every question in a similar manner. 5. What will 8 yards of muslin cost at 9 cents a yard ? 6. What will 9 pounds of sugar cost at 9 cents a pound ? 7. What is the cost of 7 pounds of butter at 12 cents a pound ? 8. What is the cost of 12 pounds of tea at 6 shillings a pound ? 9. What is the cosf of 12 pounds of coffee at 9 cents a pound ? 10. What is the cost of 11 yards of cloth at 6 dollars a yard ? 11. What is the cost of 9 books at 11 cents each ? 44 MULTIPLICATION. 12. What is the cost of 12 pencils at 8 cents apiece ? 13. What is the cost of 10 pairs of shoes' at 2 dollars a pair ? 14. What is the cost of 12 pairs of stockings at 3 shillings a pair ? PRINCIPLES AND EXAMPLES 45. Let it bo required to multiply 4 by 3, and also to mul- tiply 5 by 3. OPERATION. li -t-3 -3 i i i 4 X3 = 1 4 12 Product. OPERATION. 15 Product. From the first of these examples we see, that the product of 4 multiplied by 3, is 12, the number which arises from taking 4, 3 times ; and that the product of 5 by 3 is equal to 15, the number which arises from taking 5, three times : hence, MULTIPLICATION is the operation of taking one number as many times as there are units in another. The number to be taken is called the multiplicand. The number denoting how many times the multiplicand is taken, is called the multiplier. The result of the operation is called the product. The multiplicand and multiplier are called factors, or pro- ducers of the product. 46. We also see, from the above examples, that 4 taken 3 times, gives the same result as is obtained by adding three 4's together ; and that 5 taken 3 times gives the same result as is obtained by adding three 5's together : hence, 45. What is Multiplication ? What is the number called which is to be taken? What does the multiplier denote? What is the result called ? What are the multiplier and multiplicand called ? 46. What is 4 multiplied by 3 equal to ? What is 5 multiplied by 3 equal to ? How then may multiplication be di -lined ? SIMPLE NUMBERS. 45 MULTIPLICATION is a short method of addition. 47. The sign x, placed between two numbers, denotes that they are to be multiplied together. It is called the sign of multiplication. Also, ( 4 -f 3 ) x 5, denotes that the sum of 4 and 3 is to be multiplied by 5. 9x8= 72. Ix2x 3= 6. Ix4x 5= 20. 2x6x 5= 60. 3 x 4 x 9 = how many ? 4x3x11= how many ? 5 x 2 x 9 = how many ? 6 x 2 x 5 = how many ? 7 x 8 = how many ? 1 x 6 x 9 = how many ? 1 x 9 x 12= how many ? 5 x 2 x 11= how many ? 7 x 1 x 12= how many ? 9 x 1 x 9= how many ? 11 x 1 x 7 = how many ? 12 x 1 x 5= how many ? NOTE. There are three parts in every operation of multiplica- tion. First, the multiplicand: second, the multiplier: and third, the product. 48. The product of two factors is the same, whichever be taken for the multiplier. / ( For, let it be required to multiply 5 by 3. OPERATION. ANALYSIS. Place as many 1's in a ,5 horizontal row as there are units in the , multiplicand, and make as many rows as Mill! there are units in the multiplier : the \ | product is equal to the number of 1's in o -j 1 one row taken as many times as there are ( 1 11 1 1 rows : that is, to x 3=15. JT But if we consider the number of 1 s in a vertical row to be the multiplicand, and the number of vertical rows the multiplier, the product will be equal to the number of 1's in a vertical row taken as many times as there are vertical rows ; that is, 3 x 5=15 : and, as the same may be shown for any two numbers, The product of two factors is the same whichever factor is used as the multiplier. 47. What is the sign of multiplication ? NOTE. How many parts are there in any operation of multiplica- tion ? What are they ? 48. What is the product of 3 by 4 ? Of 4 by 3 ? Is the product altered by changing the order of the factors ? 4:6 MULTIPLICATION. EXAMPLES. 3x7 = 7x3 = 21: also, 6x3 = 3x6=18. 9 x 5=5 x 9=45 : also, 8 x 6=6 x 8 = 48. and, 8x7 = 7x8=56: also, 5x7 = 7x5 = 35. - 49. When the multiplier does not exceed 12 1. Let it be required to multiply 236 by 4. ANALYSIS. It is required to take 230 4 OPERATION. times. If the entire number is taken 4 times, 236 each order of units must be taken 4 times : 4. hence, the product must contain 24 units, 12 - tens, and 8 hundreds ; therefore, the product 24 units. is 944. 12 tens. It is seen, from the preceding analysis. 8 _ hundreds. that, 944" Product. 1. If units be multiplied by units, the unit of the product will be 1. 2. If tens be multiplied by units, the unit of the product unit be 1 ten. 3. If hundreds be multiplied by units, the unit of the product will be 1 hundred ; and so on : And since the product of the factors is the same whichever is taken for the multiplier (Art. 48), it follows that, 4. If units of the first order be multiplied by units of a higher order, the units of the product will be the mme as that of the higher order. / The operation in the last example may be performed ia another way, thus : ANALYSIS. Say 4 times 6 are 24 : set down the OPERATION. 4, and then say, 4 times 3 are 12, and 2 to carry 236 are 14 ; set down the 4, and then say, 4 times 2 are 4 8, and 1 to carry are 9. Set down the 9, and the product is 944 as before. The method of carrying is the same as in addition. (1.) (2.) (3.) (4.) 867901 278904 678741 3021945 1 2 . 3 _J 867901 12087780 SIMPLE NUMBERS. 47 (5.) (6) (7.) (8.) 28432 82798 6789 49604 8 _ _9 11 _ 12 227456 595248 9. A merchant sold 104 yards of cotton sheeting at 9 cents a yard : what did he receive for it ? 10. A farmer sold 309 sheep at four dollars apiece : how much did he receive ? 11. Mrs. Simpkins purchased 149 yards of table linen at two dollars a yard : how much did she pay for it ? 12. What is the cost of 2974 pine-apples at 12 cents apiece ? 13. What is the cost of 4073 yards of cloth at 7 dollars a yard ? 14. What is the cost of a drove of 598 hogs at 11 dollars apiece ? READING RESULTS. 50. Spelling, IP multiplication, is naming the two factors which produce the product, as well as the words which in- dicate the operation ; whilst the reading consists in naming only the word which expresses the final result. ANALYSIS. In multiplying 8325 by 6, we say, OPERATION. 6 times 5 are 30 ; then, 6 times 2 are 12 and 3 to 8325 carry are 15 ; 6 times 3 are 18 and 1 to carry are 6 19 ; C times 8 are 48 and 1 to carry are 49. This is the spelling. The reading consists in pronouncing only each final word which denotes the result of an operation thus : thirty, fifteen, nineteen, forty-nine. With a little practice, the pupils will perform the operations mentally, and read with great facility, either separately or in concert in classes. 51. When the multiplier exceeds 12. i. Multiply 8204 by 603. 49. Explain the multiplication of 336 by 4 ? What principles are established by this operation ? 50. Explain the manner of reading the results in the operations of multiplication ? 51. Give the rule for multiplication 48 MULTIPLICATION. ANALYSIS. The multiplicand is to be taken 603 R90 1 times. Taking it 3 times we obtain 24612. When we come to take it 6 hundreds times, the _ 5__ lowest order of units will be hundreds: hence, 4, 24612 the first figure of the product, must be written in 10091 the third place. 4947012 NOTE. The product obtained by multiplying by a single figure of the multiplier, is called a partial product. In the above ex- ample there are two partial products, 24612 and 49224. The sum of the partial products is equal to the result or product sought : hence, the following RULE I. Write the multiplier under the 'multiplicand, placing units of the same order in the same column. II. Beginning ivith the units' figure, multiply the entire multiplicand by each figure of the multiplier, observing to write the first figure of each partial product directly under its multiplier. , III. Add the partial products and their sum will be the product sought. PROOF. 52. Write the multiplicand in the place of the multiplier and find the product as before. If the two products are the same, the work is supposed to be right. NOTE. This proof depends on the principle that the product of two numbers is the same whichever is taken for the multiplicand (Art. 48) ; and also on the fact, that the same error would not be likely to occur in both operations. EXAMPLES. 1. Multiply 354 by 267. Multiplicand, Multiplier, Product, OPERATION. 354 267 "2478 2124 708 PROOF. 267 354 1068 1335 801 94518 94518 52. How do you prove multiplication ? SIMPLE NUMBERS. 2. Multiply 365 by 84 ; also 37864 by 209. (2.) Multiplicand, 365 Multiplier, 84 (3.) 37864 209 (4.) 34293 74 (5.) 47042 91 1460 2920 Product, 30660 4280822 (6.) 46834 679084 (8.) 1098731 (9.) 8971432 406 126 1987 10471 19014604 10. Multiply 12345678 by 32. 11. Multiply 9378964 y 42. 12. Multiply 1345894 by 49. 13. Multiply 576784 by 64. 14. Multiply 596875 by 144. 15. Multiply 46123101 by 72. 16. Multiply 6185720 by 132. 17. Multiply 7 18328 by 96. 18. Multiply five thousand nine hundred and si^ty-five, by six thousand and nine. 19. Multiply eight hundred and seventy thousand six hun- dred and fifty-one, by three hundred and seven thousand and four. 20. Multiply four hundred and sixty-two thousand six hun- dred and nine, by itself. 21. Multiply eight hundred and forty-nine million, six hun- dred and seven thousand, three hundred and six, by nine hundred thousand, two hundred and four. 22. Multiply 679534 by 9185. 23. Multiply 86972 by 1208. 24. Multiply 1055054 by 570. 25. Multiply 538362 by 9258. 26. Multiply 50406 by 8050. 27. Multiply 523972 by 1527. 28. Multiply 760184 by 1615. 29. Multiply 105070 by 3145. CONTRACTIONS IN MULTIPLICATION. 53. Contractions in multiplication are short methods of finding the product when the multiplier is a composite num- ber. 53. What are contractions in multiplication ? 4 50 MULTIPLICATION. CASE I. Of Components or Factors. 54. A composite number is one that may be produced by the multiplication of two or more numbers, which are called components or factors. Thus, 2 x 3=6. Hence, 6 is the composite number, and 2 and 3 are its components or factors. The number, 16=8x2: here 16 is a composite number, and 8 and 2 are the factors. But since 4 x4=16, we may also regard 4 and 4 as factors of 16. Again, 16=8x2, and 8 = 4x9 = 2x2x2: hence, 16=2x2x2x2: therefore, 16 has also four equal factors. 1. What are the factors of 8 ? of 9 ? of 10 ? of 12? of 14? of 18 ? of 24 ? 2. What are the factors of 20 ? of 21 ? of 22 ? of 26 ; of 25? of 30? 3. What are the factors of 36 ? of 42 ? of 44 ? of 49 ? of 56? of 64? of 72? 4. Let it be required to multiply 8 by the composite num- ber 6, of which the factors are 2 and 3. 1 1 1 1 1 1 1 1(0 V Q 1* 1111111 l| 2X8=:1 * 1 1 1 1 1 1 1 * ' 50 | q (1 1 1 1 1 1 1 l|2 48 24 ' -h 1 1 1 1 1 1 1) 9 2 (11111111) 48 If we write 6 horizontal lines with 8 units in each, it is evident that the product of 8 x 6=48 will express the num- ber of units in all the lines. Let us first connect the lines in sets of two each, as at the right ; the number of units in each set will then be expressed by 8 x 2=16. But there are 3 sets ; hence, the number of units in all the sets is 16 x 3 = 48. 54. What is a composite number ? Is 6 a composite number ? What are its components or factors ? What are the factors of the composite number 16 ? What are the factors of the composite number 12 ? How do you multiply when the multiplier is a composite number? SIMPLE NUMBERS 51 Again, if we divide the lines into sets of 3 each, as at the left, the' number of units in each set will be equal to 8x's=24, and since there are two sets, the whole number of units will be expressed by24x2=48. Since the product of either two of the three factors 8, 3 and 2, win be the same whichever be taken for the multiplier (48), and since the same principle will apply to that product and the other factor, as well as to any additional factor, if introduced, it follows that, The product of any number of factors will be the same in whatever order they are multiplied : hence, the following RULE. I. Separate the composite number into its factors. II. Multiply the multiplicand and the partial products by the factors, in succession, and the last product mill be the entire product sought. EXAMPLES. 1. Multiply 327 by 12. The factors of 12 are 2 and 6 ; they are also 3 and 4 ; or fhey are 3, 2 and 2. For, 2x6 = 12, 3x4 = 12, and 3x2x2 = 12. 2. Multiply 5709 by 48. 3. Multiply 342516 by 56. 4. Multiply 209402 by 72. 5. Multiply 937387 by 54. 6. Multiply 91738 by 81. 7. Multiply 3842 by 144. CASE II. 55. When the multiplier is 1, with any number of ci- phers annexed, as 10, 100, 1000, &c. Placing a cipher on the right of a number, is called an- nexing it. Annexing one cipher increases the unit of each place ten times : that is, it changes units into tens, tens into hundreds, hundreds into thousands, &c. ; and therefore in- creases the number ten times. Thus, the number 5 is increased ten times by annexing one cipher, which makes it 50. The annexing of two ciphers 55. If yon place one cipher on the right of a number, what effect has it on its value ? If you place two, what effect has it ? If you place three ? How much will each increase it ? How do you multiply by 10, 100, 1000, &c ? 52 MULTIPLICATION. increases a number one hundred times ; the annexing of three ciphers, a thousand times, &c. : hence the following RULE. Annex to the multiplicand as many ciphers as there are in the multiplier, and the number so formed will be the required product. EXAMPLES. 1. Multiply 254 by 10. 2. Multiply 648 by 100. 3. Multiply 7987 by 1000. 4. Multiply 9840 by 10000. 5. Multiply 3750 by 100. 6. Multiply 6704 by 10000. 7. Multiply 2141 by 100. 8. Multiply 872 by 100000. CASE III. 56. When there are ciphers on the right hand of one or both of the factors. In this case each number may be regarded as a composite number, of which the significant figures are one factor, and 1, with the requisite number of ciphers annexed, the other. 1. Let it be required to multiply 3200 by 800- OPERATION. 3200=32 x 100 ; and 800=8 x 100 ; Then, 3200 x 800 = 32 x 100 x 8 x 100 = 32x8x100x100 = 2560000. Hence, we have the following RULE. Omit the ciphers and multiply the significant figures : then place as many ciphers at the right hand of the product as there are in both factors. EXAMPLES. (1.) (2.) (3.) 76400 7532000 416000 24 580 357000 133600 148512000000 4. 4871000x270000. 5. 296200x875000. 6. 3456789x567090. 7. 21200x70. 8. 359260x304000. 9. 7496430x695000. SIMPLE NUMBERS. 53 APPLICATIONS IN MULTIPLICATION. 57. The analysis of a practical question, in Multiplication, requires that the multiplier be an abstract number ; and then the unit of the product will be the same as the unit of the multiplicand. Thus, what will 5 yards of cloth cost at 7 dollars a yard ? ANALYSIS. Five yards of cloth will cost 5 times as much as 1 yard. Since 1 yard of cloth costs 7 dollars, 5 yards will cost 5 times 7 dollars, which are 35 dollars. The cost of any number of things is equal to the price of a single thing multiplied by the number. But we have seen that the product of two numbers will be the same, (that is, will contain the same number of units) whichever be taken for the multiplicand (Art. 48). Hence, in practice, we may multiply the two factors together, taking either for the multiplier, and than assign the proper unit to the product, We generally take the least number for the multiplier. QUESTIONS FOR PRACTICE. 1. There are ten bags of coffee, each containing 48 pounds : how much coffee is there in all the bags ? 2. There are 20 pieces of cloth, each containing 37 yards, and 49 other pieces, each containing 75 yards : how many yards of cloth are there in all the pieces ? 3. There are 24 hours in a day, and 7 days in a week : how many hours in a week ? 4. A merchant buys a piece of cloth containing 97 yards, at 3 dollars a yard : what does the piece cost him ? 5. A farmer bought a farm containing 10 fields ; three of the fields contained 9 acres each ; three other of the fields 12 acres each ; and the remaining 4 fields each 15 acres : how many acres were there in the farm, and how much did the whole cost at 18 dollars an acre? 6. Suppose a man were to travel 32 miles a day : how far would he travel in 365 days ? 56. When there are ciphers on the right hand of one or both the fac- tors, how do you multiply ? 57. What does the analysis of a practical question require? How do you find the cost of a single thing ? How may it be done in practice ? 54 MULTIPLICATION. 7. A merchant bought 49 hogsheads of molasses, each containing 63 gallons : how many gallons of molasses were there in the parcel ? 8. In a certain city there are 3751 houses. If each house on an average contains 5 persons, how many inhabitants are there in the city ? 9. If a regiment of soldiers contains 1128 men, how many men are there in an army of 106 regiments ? 10. If 786 yards of cloth can be made in one day, how many yards can be made in 1252 days ? 11. If 30009 cents are paid for one man's labor on a rail- road for 1 year, how many cents would be paid to 814 men, each man receiving the same wages ? 12. There are 320 rods in a mile; how many rods are there in the distance from St. Louis to New Orleans, wind. is 1092 miles ? 13. Suppose a book to contain 470 pages, 45 lines on each page, and 50 letters in each line : how many letters in the book? 14. Supposing a crew of 250 men to have provisions for 30 days, allowing each man 20 ounces a day : how many ounces have they ? 15. There are 350 rows of trees in a large orchard, 125 trees in each row, and 3000 apples on each tree : how man} 1 apples in the orchard ? 16. What is the cost of 7585 barrels of flour at 7 dollars a barrel ? 17. If a railroad car goes 27 miles an hour, how far will it run in 3 days, running 20 hours each day ? How far would it run if its rate were 37 miles an hour ? 18. If 1327 barrels of flour will feed the inhabitants of a city for 1 day, how many barrels will supply them for 2 years ? 19. A regiment of men contains 10 companies, each com- pany 8 platoons, and each platoon 34 men : how many men in the regiment ? 20. Two persons start from the same place and travel in the same direction : one travels at the rate of 6 miles an hour, the other at the rate of 9 miles an hour. If they travel 8 hours a day, how far will they be apart at the end of 17 days ? How far if they travel in opposite directions ? SIMPLE NUMBERS. 55 21. The Erie railroad is about 425 miles long, and cost 65 thousand dollars a mile : what was the entire cost of con- struction ? 22. A drover bought 106 oxen at 35 dollars a head ; it cost him 6 dollars a head to get them to market, where he sold them at 47 dollars ; did he make or lose, and how much ? 23. The great Illinois Central Railroad reaches from Chicago to the mouth of the .Ohio river, 815 miles : it cost 23500 dollars a mile : what was its entire cost ? 24. Mr. Denning's orchard is square and contains 36 trees in a row : each tree yields 4 barrels of apples which he sells for 2 dollars a barrel : how much does he get for his crop ? BILLS OF PARCELS. 58. When a person sells goods he generally gives with them a bill, showing the amount charged for them, and acknowledging the receipt of the money paid ; such bills are called Mills of Parcels. New York, Oct. 1, 1854. 25 James Johnson, Bought of W. Smith. 4 Chests of tea, of 45 pounds each, at 1 doll, a pound. 3 Firkins of butter at 1 7 dolls, per firkiu 4 Boxes of raisins at 3 dolls, per box ... 36 Bags of coffee at 16 dolls, each 14 Hogsheads of molasses at 28 dolls, each - Amount, dollars. Received the amount in full. W. Smith Hartford, Nov. 1, 1854. 26 James Hughes, Bought of W. Jones. 27 Bags of coffee at 14 dollars per bag - 18 Chests of tea at 25 dolls, per chest - 75 Barrels of shad at 9 dolls, per barrel 87 Barrels of mackerel at 8 dolls, per barrel - 67 Cheeses at 2 dolls, each - 59 Hogsheads of molasses at 29 dolls, per hogshead, Amount, dollars. Received the amount in full, for W. Jones, per James Cross. 58. What are bills of parcels ? 56 DIVISION. DIVISION. 59. 1. How many 1's are there in 1 ? How many in 2 ? In 3 ? In 4 ? In 5 ? 2. How many 2's are there in 2 ? 2 in 2 how many times ? 2 in 4 how many times ? 2 in 6 how many times ? In 8 ? 3, How many 3's in 6 ? 3 in 6 how many times ? 3 in 9? 3 in 12? 3 in 15? 3 in 18 ? DIVISION TABLE. 1 in 1 1 time 1 in 2 2 times 1 in 3 3 times 1 in 4 4 times 1 in 5 5 times 1 in 6 6 times 1 in 7 7 times 1 in 8 8 times 1 in 9 9 times 5 in 5 1 time 5 in 10 2 times 5 in 15 3 times 5 in 20 4 times 5 in 25 5 times 5 in 30 6 times 5 in 35 7 times 5 in 40 8 times 5 in 45 9 times 9 in 91 time 9 in 18 2 times 9 in 27 3 times 9 in 36 4 times 9 in 45 5 times 9 in 54 6 times 9 in 63 7 times 9 in 72 8 times 9 in 81 9 times 2 in 2 1 time 2 in 4 2 times 2 in 6 3 times 2 in 8 4 times 2 in 10 5 times 2 in 12 6 times 2 in 14 7 times 2 in 16 8 times 2 in 18 9 times 6 in 6 1 time 6 in 12 2 times 6 in 18 3 times 6 in 24 4 times 6 in 30 5 times 6 in 36 6 times 6 in 42 7 times 6 in 48 8 times 6 in 54 9 times 10 in 10 1 time 10 in 20 2 times JO in 30 3 times 10 in 40 4 times 10 in 50 5 times 10 in 60 6 times 10 in 70 7 times 10 in 80 8 times 10 in 90 9 times 3 in 3 1 time 3 in 6 2 times 3 in 9 3 times 3 in 12 4 times 3 in 15 5 times 3 in 18 6 times 3 in 21 7 times 3 in 24 8 times 3 in 27 9 times 7 in 7 1 time 7 in 14 2 times 7 in 21 3 times 7 in 28 4 times 7 in 35 5 times 7 in 42 6 times 7 in 49 7 times 7 in 56 8 times 7 in 63 9 times 11 in 11 1 time 11 in 22 2 times 11 in 33 3 times 11 in 44 4 times 11 in 55 5 times 11 in 66 6 times 11 in 77 7 times 11 in 88 8 times 11 in 99 9 times 4 in 41 time 4 in 8 2 times 4 in 12 3 times 4 in 16 4 times 4 in 20 5 times 4 in 24 6 times 4 in 28 7 times 4 in 32 8 times 4 in 36 9 times 8 in 8 1 time 8 in 16 2 times 8 in 24 3 times 8 in 32 4 times 8 in 40 5 times 8 in 48 6 times 8 in 56 7 times 8 in 64 8 times 8 in 72 9 times 12 in 12 *1 time 12 in 24 2 times 12 in 36 3 times 12 in 48 4 times 12 in 60 5 times 12 in 72 6 times 12 in 84 7 times 12 in 96 8 times 12 in 108 9 times SIMPLE NUMBERS. 57 QUESTIONS. 1. If 12 apples be equally divided among 4 boys, how many will each have ? ANALYSIS. Since 12 apples are to be divided equally among 4 boys, one boy will have as many apples as 4 is contained times in 12, which is 3. 2. If 24 peaches be equally divided among 6 boys, how many will each have ? How many times is 6 contained in 24? 3. A man has 32 miles to walk, and can travel 4 miles an hour, how many hours will it take him ? 4. How many yards of cloth, at 3 dollars a yard, can you buy for 24 dollars ? ANALYSIS. Since the cloth is 3 dollars a yard, you can buy as many yards as 3 is contained times in 24, which is 8 : therefore, you can buy 8 yards. 5. How many oranges at 6 cents apiece can you buy for 42 cents ? 6. How many pine-apples at 12 cents apiece can you buy for 132 cents ? 7. A farmer pays 28 dollars for 7 sheep : how much is that apiece ? ANALYSIS. Since 7 sheep cost 28 dollars, one sheep will cost as many dollars as 7 is contained times in 28, which is 4 ; therefore, each sheep will cost 4 dollars. 8. If 12 yards of muslin cost 96 cents, how much does 1 yard cost ? 9. How many lead pencils could you buy for 42 cents, if they cost 6 cents apiece ? 10. How many oranges could you buy for 72 cents, if they cost 6 cents apiece ? 11. A trader wishes to pack 64 hats in boxes, and can put but 8 hats in a box : how many boxes does he want ? 12. If a man can build 7 rods of fence in a day, how long will it take him to build 7 7 rods ? 13. If a man pays 56 dollars for seven yards of cloth, how much is that a yard ? 58 DIVISION. 14. Twelve men receive 108 dollars for doing a piece of work : how much does each one receive ? 15. A merchant has 144 dollars with which he is going to buy cloth at 12 dollars a yard ; how many yards can he pur- chase ? 16. James is to learn forty-two verses of Scripture in a week : how many must he learn each day ? 17. How many times is 4 contained in 50, and how many over? PRINCIPLES AND EXAMPLES. 60. 1. Let it be required to divide 86 by 2. Set down the number to be divided and write the other number on the left, drawing a curved line between them. Now there are 8 tens and 6 units to be divided by 2. We say, 2 in 8, 4 times, which being tens, we write it in the tens' place. We then say, 2 in 6, 3 times, which being units, are written in the units' place. The result, which is called a quotient, is there- fore, 4 tens and 3 units, or 43. 2. Let it be required to divide 729 by 3. OPERATION. 2) 86 43 quotie't. ANALYSIS. We say, 3 in 7, 2 times and 1 over. OPERATION. Set down the 2, which are hundreds, under the 7. But of the 7 hundreds there is 1 hundred, or 10 tens, not yet divided. We put the 10 tens with the 2 3)729 1243 tens, making 12 tens, and then say, 3 in 12, 4 times, and write the 4 of the quotient in the tens' place ; then say, 3 in 9, 3 times. The quotient, therefore, is 243. 3. Let it be required to divide 466 by 8. ANALYSIS. We first divide the 46 tens by 8, giving a quotient of 5 tens, and 6 tens over. These 6 tens are equal to 60 units, to which we add the 6 in the units' place. We then say, 8 in 66, 8 times and 2 over ; hence, the quotient is 58, and 2 over, which we caU a remainder. This remainder is written after the last quotient figure, and the 8 paced under it; the quotient is read, 58 and 2 divided by 8- OPERATION. 8)466 58-2 remain. 58f quotient. 50. Ex. 1. When you divide 8 tons* by 2, is the unit of the quotient tens or units ? When 6 units are divided by 2, what is the unit ? SIMPLE NUMBERS. 59 ANALYSIS. In the first example 86 is divided into 2 equal parts, and the quotient 43 is one of the parts. If one of the equal parts be multiplied by the number of parts 2, the product will be 86, the number divided. In the third example 466 is divided into 8 equal parts, and two units remain that are not divided. If one of the equal parts 58, be multiplied by the number of parts, 8, and the remainder 2 be added to the product, the result will be equal to 466, the number divided. 61. DIVISION is the operation of dividing a number into two equal parts ; or, of finding how many times one number contains another. The first number, or number by which we divide, is called the divisor. The second number, or number to be divided, is called the dividend. The third number, or result, is called the quotient The quotient shows how many times the dividend contains the divisor. If anything is left after division, it is called a remainder. 62. There are three parts in every division, and sometimes four : 1st, the dividend ; 2d, the divisor ; 3d, the quotient ; and 4th, the remainder. There are three signs used to denote division ; they are the following : lS-f-4 expresses that 18 is to be divided by 4. -^ 8 expresses that 18 is to be divided by 4. 4)18 expresses that 18 is to be divided by 4. When the last sign is used, if the divisor does not exceed 12, we draw a line beneath, and set the quotient under it. If the divisor exceeds 12, we draw a curved line on the right of the dividend, and set the quotient at the right. 2. When the seven hundreds are divided by 3, what is the unit of the quotient? To how many tens is the undivided hundred equal? When the 13 tens arc divided by 8, what is the unit of the quotient? Whun the 9 uuits arc divided by #, what is the quotient ? --How is the division of the remainder expressed ? Read the quotient. If there be a remainder after division, how must it be written ? 61. What is division ? What is the number to be divided called ? What is the number called by which we divide? What is the answer called ? What is the number oalled which is left ? 62. Plow many parts arc there in division ? Name them. How many signs are there in division ? Make and name them ? 60 SHORT DIVISION. SHORT DIVISION. 63. SHORT DIVISION is the operation of dividing when the work is performed mentally, and the results only written down. It is limited to the cases in which the divisors do not exceed 12. Let it be required to divide 30456 by 8. ANALYSIS We first say, 8 in 3 we cannot. Then, OPERATION. 8 in 30, 3 times and 6 over; then 8 in 64, 8 times ; 8)30456 then 8 in 5, times; then, 8 in 50. 7 times: hence, / ooOT RULE I. Write the divisor on the left of the dividend. Beginning at the left, divide each figure of the dividend by the divisor, and set each quotient figure under its dividend II. If there is a remainder, after any division, annex (o it the next figure of the dividend, and divide as hcfnrp , ^ III. Jf any dividend is less than the divisor, write 0/br the quotient figure and annex the next figure of the dividend, for a new dividend. IV. If there is a remainder, after dividing the last figure, set the divisor under it, and annex the result to the quotient. PROOF. Multiply the divisor by the quotient, and to the product add the remainder, when there is one ; if the work is right the result will be equal to the dividend. / EXAMPLES. (1.) (2.) (3,) (4) 3)9369 4)73684 5)673420 6)825467 Ans. 3123 18421 134684 137577f 3 4 5 6_ Proof 9369 73684 673420 825467" 5. Divide 86434 by 2. 6. Divide 416710 by 4. 7 Divide 641 40 by 5. 8. Divide 278943 by 6. 9. Divide 95040522 by 6. 10. Divide 75890496 by 8. 11. Divide 6794108 by 3. 12. Divide 21090431 by 9. 13. Divide 2345678964 by 6 14 Divide 570196382 by 12 15. Divide 67897634 by 9. 16. Divide 75436298 by 12. 17. Divide 674189904 by 9. 18. Divide 1404967214 by 11. 19. Divide 27478041 by 10 20 Divide 167484329 by 12. EQUAL PARTS. 61 21. A man sold his farm for 6756 dollars, and divided the amount equally between his wife and 5 children : how much did each receive ? 22. There are 576 persons in a train of 12 cars : how many are there in each car ? 23. If a township of land containing 2304 acres be equally divided among 8 persons, how many acres will each have ? 24. If it takes 5 bushels of wheat to make a barral of flour, how many barrels can be made from 65890 bushels ? 25. Twelve things make a dozen : how many dozens are therein 2167284? 26. Eleven persons are all of the same age, and the sum of their ages is 968 years : what is the age of each ? 27. How many barrels of flour at 7 dollars a barrel can be bought for 609463 dollars ? 28. An estate worth 2943 dollars, is to be divided equally among a father, mother, 3 daughters and 4 sons : what is the portion of each ? 29. A county contains 207360 acres of land lying in 9 town- ships of equal extent : how many acres in a township ? 30. If 11 cities contain an equal number of inhabitants, and the whole number is equal to 3800247 : how many will there be in each ? EQUAL PARTS OF NUMBERS. 64. 1. If any number or thing be divided into two equal parts, one of the parts is called one-half: one half of a single thing is written thus ; J. 2. If any number is divided into three equal parts, one of the parts is called one-third, which is written thus ; \ ; two of the parts are called two-thirds: which are written thus ; f . 3. If any number is divided into four equal parts, one of the parts is called one-fourth, which is written thus ; J ; two of the parts are called two-fourths, and are written thus ; ; three of them are called three-fourths, and written J ; and similar names are given to the equal parts into which any number may be divided. 63. What is short division ? How is it generally performed ? Give the rule ? How do you prove short division ? 62 EQUAL PARTS 4. If a number is divided into five equal parts, what is one of the parts called ? Two of them ? Three of them ? Pour of them ? 5. If a number is divided into 7 equal parts, what is one of the parts called ? What is one of the parts called when it is divided into 8 equal parts ? When it is divided into 9 equal parts ? When it is divided into 10 ? When it is divided into 11 ? When it is divided into 12 ? - 6. What is one-half of 2? of4? of6? ofS? of 10? of 12? of 14? of 16? of 18? 7. What is two-thirds of 3 ? ANALYSTS Two-thirds of three are two times one third of three. ODe-third of three is 1 , therefore, two-thirds of three are two times 1, or 2. Let every question be analyzed in the same manner. What is one-third of 6 ? 2 thirds of 6 ? One-third of 9 ? 2 thirds of 9 ? One-third of 12 ? two-thirds of 12 ? 8. What is one-fourth of 4 ? 2 fourths of 4 ? 3 fourths of 4 ? What is one-fourth of 8 ? 2 fourths of 8 ? 3 fourths of 8 ? What is one-fourth of 12 ? 2 fourths of 12 ? 3 fourths of 12 ? One- fourth of 16 ? 2 fourths of 16 ? 3 fourths ? 9. What is one-seventh of 7 ? What is 2 sevenths of 7 ? 5 sevenths? 6 sevenths? What is one-seventh of 14? 3 sev- enths ? 5 sevenths ? 6 sevenths ? What is one-seventh of 21 ? of 28 ? of 35 ? 10. What is one-eighth of 8? of 16? of 24? of 32? of 40? of 56? 1 1 . What is one-ninth of 9 ? 2 ninths ? 7 ninths ? 6 ninths ? 5 ninths? 4 ninths? What is one-ninth of 18? of 27? of 54? of 72? of 90? of 108? 12. How many halves of 1 are there in 2 ? ANALYSIS There are twice as many halves in 2 as there are in 1. There are two halves in 1 ; therefore, there are 2 times 2 ''halves in 2, or 4 halves. 13 How many halves of 1 are there in 3 ? In 4 ? In 5 ? In 6? In 8? In 10? In 12? 14 How many thirds are there in 1 ? How many thirds of 1 in 2? In 3? In 4? In 5? In 6? In 9? In 12? 15. How many fourths are there in 1 ? How many fourths of 1 in 2? In 4? In 6? In 10? In 12? OF NUMBERS. 16. How many fifths are there in 1 ? How many fifths of 1 are there in 2 ? In 3 ? In 6 ? In 1 ? In 11 ? In 12 ? 17. How many sixths are there in 2 and one-sixth ? In 3 and 4 sixths ? In 5 and 2 sixths ? In 8 and 5 sixths ? 18. How many sevenths of 1 are there in 2 ? In 4 and 3 sevenths how many ? How many in 5 and 5 sevenths ? In fc 5 and 6 sevenths ? 19. How many eighths of 1 are there in 2 ? How many in 2 and 3 eighths ? In 2 and 5 eighths ? In 2 and 7 eighths? In 3 ? In 3 and 4 eighths ? In 9 ? In 9 and 5 eighths ? In 10 ? In 10 and 7 eighths ? 20. How many twelfths of 1 are there in 2 ? In 2 and 4 twelfths how many ? How many in 4 and 9 twelfths ? How many in 5 and 10 twelfths? In 6 and 9 twelfths? In 10 and 11 twelfths? 21. What is the product of 12 multiplied by 3 and one half, (which is written 3J) ? ANALYSIS. Twelve is to be taken 3 and one-half times (Art 45). Twelve taken times is 6 ; and 12 taken three times is 36 ; therefore, 12 taken ty times is 42. 22. What is the product of 10 multiplied by 5J ? 23. What is the product of 12 multiplied by 3J ? 24. What is the product of 8 multiplied by 4 J ? 25. What will 9 barrels of sugar cost at 2 dollars a barrel? ANALYSIS. Nine barrels of sugar will cost nine times as much as 1 barrel. If one barrel of sugar costs 2f dollars, 9 barrels will cost 9 times 2f dollars, which are 24 dollars. For, 2 thirds taken 9 times gives 18 thirds, which are equal to 6 ; then 9 times 2 are 18, and 6 added gives 24 dollars. 26. What will 6 yards of cloth cost at 5 dollars a yard ? 27. What will 12 sheep cost at.4J dollars apiece ? 28. What will 10 yards of calico cost at 9f cents a yard ? 29. What will 8 yards of broadcloth cost at 7-J dollars a yard ? / - 30. What will 9 tons of hay cost at 9^ dollars a ton ? 31. How many times is 2J contained in 10 ? ANALYSIS. Two and one-half is equal to 5 halves ; and 10 is equal to 20 halves ; then 5 halves is contained in 20 halves 4 times: hence. LONG DIVISION. In all similar questions change the divisor and dividend to the same fractional unit. (Art. 144). 32. How many yards of cloth, at 3J dollars a yard, can you buy for 14 dollars ? how many for 21 dollars ? 33. If oranges are 3| cents apiece, how many can you buy for 20 cents ? ' , : 34. If 1 yard of nbbon costs 2f cents, how many yards can you buy for 12 cents ? 35. If 1 yard 'of broadcloth costs 3| dollars, how many- yards can be bought for 33 dollars ? 36. If 1 pound of sugar costs 4J cents, how many pounds can be bought for 36 cents ? / 37. How many times is 5J contained in 44 ? 38. How many times is 2| contained in 24 ? 39. How many lemons, at 2| cents apiece, can you buy for 32 cents ? 40. How many yards of ribbon, at 1^ cents a yard, can you buy for 12 cents ? LONG DIVISION. 65. LONG DIVISION is the operation of finding the quotient of one number divided by another, and embraces the case of Short Division, treated in Art. 63. 1. Let it be required to divide 7059 by 13. ANALYSIS. The divisor, 13, is not OPERATION. contained in 7 thousands ; therefore, . ^ there are no thousands in the quotient. & ^ J J& ' m -3 We then consider the to be annex- J2 s g '3 a g *3 ed to the 7, making 70 hundreds, and EH W EH P W EH P call this a partial dividend. 13)70 5 9(5 43 The divisor, 13, is contained in 70 65 hundreds, 5 hundreds times and some- ^-- thing over. To find how much over, multiply 13 by 5 hundreds and subtract 5 2 the product 65 from 70, and there will r 3 g remain 5 hundreds, to which bring q down the 5 tens and consider the 55 _r__ tens a new partial dividend. 65. What is long division ? Does it embrace the case of short divi- sion ? What is u partial dividend ? SIMPLE NUMBERS. 65 Then, 13 is contained in 55 tens, 4 tens times and something over. Multiply 13 by 4 tens and subtract the product, 52, from 55, and to the remainder 3 tens bring down the 9 units, and con- sider the 39 units a new partial dividend. Then, 13 is contained in 39, 3 times. Multiply 13 by 3, and subtract the product 39 from 39, and we find that nothing remains. 66. PROOF. Each product that has arisen from multiply- ing the divisor by a figure of the quotient, is a partial product, and the sum of these products is the product of the divisor and quotient (Art. 51, XOTE). Each product has been taken, separately, from the dividend, and nothing remains. But, taking each product away in succession, leaves the same re- mainder as would be left if their sum were taken away at once. Hence, the number 543, when multiplied by the divisor, gives a product equal to the dividend : therefore, 543 is the quotient (Art. 61) : hence, to prove division, Multiply the divisor by the quotient and add in the remain- der, if any. If the work is right, the result will be the same as the dividend. 67. Let it be required to divide 2756 by 26. We first say, 26 in 27 once, and place 1 in OPERATION. the quotient. Multiplying by 1, subtracting, 26)2756(106 and bringing down the 5, we have 15 for the 26 first partial dividend. We then say, 26 in 15, "^ times, and place the in the quotient. We 156 then bring down the 6, and find that the divisor 156 is contained in 156, 6 times. If anyone of the partial dividends is less than the divisor, write for the quotient figure, and then bring down the next figure, forming a new partial dividend. Hence, for Long Division, we have the following KULE. I. Write the divisor on the left of the dividend. II. Note the fewest figures of the dividend, at the left, that will contain the divisor, and set the quotient figure at the right. 66. What is a partial product ? What is the sum of all the partial products equal to ? How do you prove division ? 67. What do you do if any partial dividend is less than the divisor ? What is the rule for long division ? 66 LONG DIVISION. III. Multiply the divisor by the quotient figure, subtract the product from the first partial dividend, and to the re- mainder annex the next figure of the dividend, forming a second partial dividend. TV. find in the same manner the second and succeeding figures of the quotient, till all the figures of the dividend are brought down. NOTE 1. There arc five operations in Long Division. 1st. To write down the numbers : 2d. Divide, or find how many times : 3d. Multiply : 4th. Subtract : 5th. Bring down, to form the partial uividends. 2. The product of a quotient figure by the divisor must never be larger than the corresponding partial dividend : if it is, the quotient figure is too large and must be diminished. 3. When any one of the remainders is greater than the divisor, the quotient figure is too small and must be increased. 4. The unit of any quotient figure is the same as that of the partial dividend from which it is obtained. The pupil should always name the unit of every quotient figure. EXAMPLES. 1. Divide 7574 by 54. OPERATION. 54)7574/140 54 2. Divide 67289 by 261. OPERATION. 261)67289(257 522 1508 1305 2039 1827 212 Remainder, PROOF. 140 Quotient. 54 Divisor. 560 700 7560 14 Remainder. 7574 Dividend. PROOF. 261 Divisor. 257 Quotient. 1827 1305 522 212 Remainder. -#7289 Dividend. SIMPLE NUMBERS. 67 3. Divide 119836687 by 39407. OPERATION. PROOF. 39407)119836687(3041 39407 Divisor. 118221 3041 Quotient. 161568 39407 157628 157628 39407 . 118221 39407 119836687 Dividend. 4. Divide 7210473 by 37. 5. Divide 147735 by 45. 6. Divide 937387 by 54. 7. Divide 145260 by 108 8. Divide 79165238 by 238. 9. Divide 62015735 by 78. 10. Divide 14420946 by 74. 11. Divide 295470 by 90. 12. Divide 1874774 by 162. 13. Divide 435780 by 216. 14. Divide 203812983 by 5049. 15. Divide 20195411808 by 3012. 16. Divide 74855092410 by 949998. 17. Divide 47254149 by 4674. 18. Divide 119184669 by 38473. 19. Divide 280208122081 by 912314. 20. Divide 293839455936 by 8405. 21. Divide 4637064283 by 57606. 22. Divide 352107193214 by 210472. 23. Divide 558001172606176724 by 2708630425. 24. Divide 1714347149347 by 57143. 25. Divide 6754371495671594 by 678957 26. Divide 71900715708 by 37149. 1 27. Divide 571943007145 by 37149. 28. Divide 671493471549375 by 47143. 29. Divide 571943007645 by 37149. 30. Divide 171493715947143 by 57007. 31. Divide 121932631112635269 by 987654321. NOTES. 1. How many operations are there in long division ? Name them. 2. If a partial product is greater than the partial dividend, what does it indicate ? What do you do ? 3. What do you do when any one of the remainders is greater than the divisor ? 4. What is the unit of any figure of the quotient ? When the divisor is contained in simple units, what will be the unit of the quotient figure ? When it is contained in tens, what will be the unit of the quotient figure ? When it is contained in hundreds ? In thousands ? 68 LONG DIVISION. 08. PRINCIPLES RESULTING FROM DIVISION. NOTES. 1st. When the divisor is 1, the quotient will be equal to the dividend. 2d. When the divisor is equal to the dividend, the quotient ' will be 1. 3d. "When the divisor is less than the dividend, the quotient will be greater than 1. The quotient will be as many times greater than 1, as the dividend is times greater than the divisor. 4th. When the divisor is greater than the dividend, the quotient will be less than 1. The qaot'ent will be such a part of 1, as the dividend is of the divisor. PROOF OF MULTIPLICATION. 69. Division is the reverse of multiplication, and they prove each other. The dividend, in division, corresponds to the product in multiplication, and the divisor and quotient to the multiplicand and multiplier, Avhich are factors of the pro- duct : hence, If the product of two numbers be divided by the multipli- cand, the quotient will be the multiplier ; or, if it be divided by the multiplier, the quotient will be the multiplicand. EXAMPLES. 3679 Multiplicand 3679J1203033(327 327 -Multiplier. 11037 25753 9933 7358 7358 11037 25753 1203033 Product. 25753 2. The multiplicand is 61835720, and the product 8162315040 : what is the multiplier ? 3. The multiplier is 270000 ; now if the product be 1315170000000, what will be the multiplicand? 4. The product is 68959488, the multiplier 96 : what is the multiplicand ? 5. The multiplier is 1440, the product 10264849920 : what is the multiplicand ? 6. The product is 6242102428164, the multiplicand 6795634 : what is the multiplier ? CONTRACTIONS IN MULTIPLICATION. G9 CONTRACTIONS IN MULTIPLICATION. 70. To multiply by 25. 1. Multiply 275 by 25.' ANALYSIS. If we annex two ciphers to the mul- OPERATION-. tiplicand, we multiply it by 100 (Art. 55): this 4)27500 product is 4 times too great ; for the multiplier is /,, 7 - but one-fourth of 100 ; hence, to multiply by 25, Annex two ciphers to the multiplicand and divide the result by 4. EXAMPLES. 1. Multiply 127 by 25. 2. Multiply 4269 by 25. 3. Multiply 87504 by 25. 4. Multiply 7-04963 by 25. 71. To multiply by 12 J 1. Multiply 326 by m. ANALYSIS. Since 12^ is one-eighth of 100, OPERATION. Annex two ciphers to the multiplicand and di- 8)32600 vide the result by 8. 4.075 EXAMPLES. 1. Multiply 284 by 12J. 2. Multiply 376 by 121. 3. Multiply 4740 by 12. 4. Multiply 70424 by 12 72. To multiply by 33* 1. Multiply 675 by 33J. ANALYSIS. Annexing two ciphers to the mul- OPERATION. tiplicand, multiplies it by 100: but the multiplier 3)67500 is but one-third of 100 : hence, Annex two ciphers and divide the result ly 3. EXAMPLES. 1. Multiply 889626 by 33J. 2 Multiply 740362 by 33J. 3. Multiply 5337756 by 33J. 4. Multiply 2221086 by 33i. 68. When the divisor is 1, what is the quotient? Wheii the divisor is equal to the dividend, what is the quotient ? When the divisor is less than the dividend, how does the quotient compare with 1 ? When the di- visor is greater than the dividend, how doas the quotient compare with 1 ? 09. If a product be divided by one of the factors, what is the quotient ? 70 CONTRACTIONS IN MULTIPLICATION. 73. To multiply by 125. 1. Multiply 375 by 125. ANALYSIS. Annexing three ciphers to the mul- tiplicand, multiplies it by 1000 : but 125 is but one-eighth of one thousand : hence, Annex three ciphers and divide the result by 8. OPERATION. 8)375000 46875 EXAMPLES. 1. Multiply 29632 by 125. 2. Multiply 8796704 by 125. 3. Multiply 970406 by 125. 4. Multiply 704294 by 125. 74. By reversing the last four processes, we have the four folio whig rules : 1. To divide any number by 25 ; Multiply the number by 4, and divide the product by 100. 2. To divide any number by 12. Multiply the number by 8, and divide the product by 100. 3. To divide any number by 33 \ : Multiply the number by 3, and divide the product by 100. 4. To divide any number by 125 : Multiply by 8, and divide the product by 1000. EXAMPLES. 1. 2. 3. 4. 6. 6. 7. 8. Divide Divide Divide Divide Divide Divide Divide Divide 3175 by 25. 106725 by 25. 2187600 by 25. 2426225 by 25. 1762405 by 25. 4075 by 12J. 3550 bv 12J. 59262$ by 12J. 9. 10. 11. 12. ,13. 14 15. 16. Divide Divide Divide Divide Divide Divide Divide Divide 880300 by 12i. 22500 by 33J. 654200 by 33J. 7925200 by 33. 4036200 by 33f . 93750 by 125. 3007875 by 125. 6758625 by 125. 70. What is the rule for multiplying by 25 ? 71. What is the rule for multiplying by 12* ? 72. What is the rule for multiplying by 88* ? 73. What is the rule for multiplying by 135? CONTRACTIONS IN DIVISION. 71 CONTRACTIONS IN DIVISION. 75. Contractions in Division are short methods of finding the quotient, when the divisors are composite numbers. CASE I. 76. When the divisor is a composite number. 1. Let it be required to divide 1407 dollars equally among 2i rnen. Here the factors of the divisor are 7 and 3. ANALYSIS. Let the 1407 dollars be first divided into 7 equal piles. OPERATION. Each pile will contain 201 dollars. 7)1407 Let each pile be now divided into 3 , .... . , , , . equal park Each part will contain S) 201 lst quotient. 67 dollars, and the number of parts G7 quotient sought, will bo 21 : hence the following RULE. Divide the dividend by one of the factors of the divisor ; tlien divide the quotient, thus arising, by a second factor, and so on, till every factor has been used as a divisor : the last Quotient will be the answer. EXAMPLES. Divide the following nnmbers by the factors ; 1. 1260 by 12 3x4. 2. 18576 by 48=4 x 12. 3. 9576 by 72 = 9x8. 4. 19296 by %=12x8. 5. 55728 by4x 9x4=14 4. 6. 92880 by 2x2x3x2x2. 7. 57888 by4x2x2x2. 8. 154368 by 3 x 2 x fc. NOTE. It often happens that there are remainders after some of the divisions How are we to find ihe true remainder? 74. 1. What is the rule for divicling by 25 ? 2. What is the rule for dividing by 12* ? 3. What is the rule for dividing by 33* ? 4. What is the rale for dividing by 125 ? 75. What are contractions in division ? What is a composite num- ber? 76. What is the rule for division when the divisor is a composite number ? 72 CONTRACTIONS. 77. Let it be required to divide 751 grapes into 16 equal parts. (4)751 4 x 4 = 16 -j 4)18T .... 3 first remainder. 40 .... 3x4 = 12 3 15 true rem. 4ns. -4S}|. NOTE. The factors of the divisor 16, are 4 and 4. ANALYSIS. If 751 grapes be divided by 4, there will be 187 bunches, each containing 4 grapes, and 8 grapes over. The unit of 187 is one bunch ; that is, a unit 4 times <(s great as 1 grape. If we divide 187 bunches by 4, we shall have 46 piles, each containing 4 bunches, and 3 bunches over : here, again, the unit of the quotient is 4 times as great as the unit of the dividend. If, now we wish to find the number of grapes not included in the 46 piles, we have 3 bunches with 4 grapes in a bunch, and 3 grapes besides : hence, 4 x 3 = 12 grapes ; and adding 3 grapes, we have a remainder, 15 grapes ; therefore, to find the remainder, in units of the given dividend : I. Multiply the last remainder by the last divisor but onr, and add in the preceding remainder : II. Multiply this result by the next preceding divisor, and add in the remainder, and so on, till you reach the unit of the dividend. EXAMPLES, 1. Let it be required to divide 43720 by 45. 3)43720 5)14573 . l = lstrem. 1x5 + 3-8; 3)2914 . 3= 3d rem. 8x3 + 1 = 25 971 . 1 = 3d rein. 25 true r era. Divide the following numbers by the factors, for the divisors : 2. 956789 by 7x8 = 56. 3. 4870029 by 8x9 = 72. 4. 674201 by*10x 11 = 110. 5. 4-15767 by 12x12 = 144. 6. 1913578 by 7x2x3 = 42. 7. 146187 by 3x5x7 = 105. 8. 26964 by 5x2 x 11 = 110. 9. 93696 by 3x7x11 = 231. 77. Give the rule for the remainder. IN DIVISION. 73 CASE II. 78. When the Divisor is 10, 100, 1000, &c. ANALYSIS. Since any number is made up of units, tens, hun- dreds, &c. (Art. 28), the number of tens in any dividend will denote how many times it contains 1 ten, and the units "will be the remainder. The hundreds will denote how many times the divi- dend contains 1 hundred, and the tens and units will be thi3 remain- der ; and similarly, when the divisor is 1000, 10000, &c. ; hence, RULE. Cut off from the right hand as many figures as there are ciphers in the divisor the figures at the left ivill be the quotient, and those at the right, the remainder. EXAMPLES. 1. Divide 49763 by 10. 2. Divide 7641200 by 100. 3. Divide 496321 by 1000. 4. Divide 6i9T8 by 10000. CASE III. 79. When there are ciphers on the right of the divisor. I. Let it be required to divide 67389 by 700. ANALYSIS. We may regard the OPERATION. divisor as a composite number, of 7|00)673[89 which the factors are 7 and 100. We first divide by 100 by striking off the 89, and then find that 7 is 189 true remain, contained in the remaining figures, " ^ns 96- 90 times, with a remainder of 1 ; this remainder we multiply by 100, and then add 89, forming the true remainder 189 : to the quotient 96, we annex 189 divided by 700, for the entire quotient : hence, the following RULE I. Cut off" the ciphers by a line, and cut off" the same number of figures from the right of the dividend. II. Divide the remaining; figures of the dividend by the remaining figures of the divisor, and annex to the remainder, if there be one, the figures cut off from the dividend : this will form the true remainder EXAMPLES. 1. Divide 8749632 by 37000. 78. How do you divide when the divisor is 1 with ciphers annexed? Give the reason of the rule. 79. How do you divide when there are ciphers on the right of the divisor ? How do you form the true remainder ? APPLICATIONS. 371000)87491632(236 74 Ans. 236JJJJJ. 17 Divide the following numbers : 2. 986327 by 210000. 3. 876000 by 6000. 4. 36599503 by 400700. 5. 5714364900 by 36500. 6. 18490700 by 73000. 7. 70807149 by 31500. APPLICATIONS. 80. Abstractly, the object of division is to find from two given numbers a third, which, multiplied by the first, will produce the second. Practically, it has three objects : 1. Knowing the number of things and their entire cost, to find the price of a single thing : 2. Knowing the entire cost of a number of things and the price of a single thing, to find the number of things : 3. To divide any number of things into a given number of equal parts. For these cases, we have from the previous principles (page 57), the following RULES. I. Divide the entire cost by the number of the things : the quotient will be the price of a single thing. II. Divide the entire cost by the price of a single thing : the quotient will be the number of things. III. Divide the whole number of things by the number of parts into which they are to be divided : the quotient will be the number in each part. QUESTIONS INVOLVING THE PREVIOUS RULES. 1. Mr. Jones died, leaving an estate worth 4500 dollars, to be divided equally between 3 daughters and 2 sons : what was the share of each ? 80. What is the object of division, abstractly? How many objects has it, practically ? Name the three objects. Give the rules for the three cases. APPLICATIONS. 75 2. What number must be multiplied by 124 to produce 40796? 3. The sum of 19125 dollars is to be distributed equally among a certain number of men, each to receive 425 dollars : how many men are to receive the money ? 4. A merchant has 5100 pounds of tea, and wishes to pack it in 60 chests : how much must he put in each chest ? 5. The product of two numbers is 51679680, and one of the factors is 615 : what is the other factor ? 6. Bought 156 barrels of flour for 1092 dollars, and sold the same for 9 dollars per barrel : how much did I gain ? 7. Mr. James has 14 calves worth 4 dollars each, 40 sheep worth 3 dollars each ; he gives them all for a horse worth 150 dollars : what does he make or lose by the bargain ? 8. Mr. Wilson sells 4 tons of hay at 12 dollars per ton, 80 bushels of wheat at 1 dollar per bushel, and takes in payment a horse worth 65 dollars, a wagon worth 40 dollars, and the rest in cash : how much money did he receive ? 9. How many pounds of coffee, worth 12 cents a pound, must be given for 368 pounds of sugar, worth 9 cents a pound ? 10. The distance around the earth is computed to be about 25000 miles : how long would it take a man to travel that distance, supposing him to travel at the rate of 35 miles a day? 11. If 600 barrels of flour cost 4800 dollars, what will 21 7 2 barrels cost? 12. If the remainder is 17, the quotient 610, and the divi- dend 45767, what is the divisor? 13. The salary of the President of the United States is 25000 dollars a year : how much can he spend daily and save of his salary 4925 dollars at the end of the year ? 14. A farmer purchased a farm for which he paid 18050 dollars. He sold 50 acres for 60 dollars an acre, and the re- mainder stood him in 50 dollars an acre : how much land did he purchase ? 15. There are 31173 verses in the Bible: how many verses must be read each day, that it may be read through in a year ? 16. A farmer wishes to exchange 250 bushels of oats at 42 cents a bushel, for flour at 7 dollars per barrel : how many barrels will he receive ? 76 APPLICATIONS. It. The owner of an estate sold 240 acres of land and had 312 acres left : how many acres had he at first ? 18. Mr. James bought of Mr. Johnson two farms, one con- taining 250 acres, for which he paid 85 dollars per acre ; the second containing 175 acres, for which he paid 70 dollars an acre ; he then sold them both for 75 dollars an acre : did he make or lose, and how much ? 19. A farmer has 279 dollars with which he wishes to buy cows at 25 dollars, sheep at 4 dollars, and pigs at 2 dollars apiece, of each an equal number : how many can he buy of each sort ? 20. The sum of two numbers is 3475, and the smaller is 1162 : what is the greater ? 21. The difference between two numbers is 1475, and the greater number is 5760 : what is the smaller ? 22. If the product of two numbers is 346712, and one of the factors is 76 : what is the other factor? 23. If the quotient is 482, and the dividend 135442 : what is the divisor ? 24. A gentleman bought a house for two thousand twenty- five dollars, and furnished it for seven hundred and six dol- lars ; he paid at one time one thousand and ten dollars, and at another time twelve hundred and seven dollars : how much remained unpaid ? 25. At a certain election the whole number of votes cast for two opposing candidates was 12672: the successful can- didate received 316 majority : how many votes did each re- ceive ? 26. Mr. Place purchased 15 cows : he sold 9 of them for 35 dollars apiece, and the remainder for 32 dollars apiece, when he found that he had lost 123 dollars : how much did he pay apiece for the cows ? 27. Mr. Gill, a drover, purchased 36 head of cattle at 64 dollars a head, and 88 sheep at 5 dollars a head ; he sold the cattle at one-quarter advance and the sheep at one-fifth ad- vance : how much did he receive for both lots ? 28. Mr. Nelson supplied his farm with 4 yoke of oxen at 93 dollars a yoke ; 4 plows at 11 dollars apiece ; 8 horses at 97 dollars each ; and agrees to pay for them in wheat at 1 dollar and a half per bushel ; how many bushels must he give ? APPLICATIONS. 77 29. If a man's salary is 800 dollars a year and his expenses 425 dollars, how many years will elapse before he will be worth 10000 dollars, if he is worth 2500 dollars at the pre- sent time ? 30. How long can 125 men subsist on an amount of food that will last 1 man 4500 days ? 31. A speculator bought 512 barrels of flour for 3584 dol- lars and sold the same for 4608 dollars : how much did he gain per barrel ? 32. A merchant bought a hogshead of molasses containing 96 gallons at 35 cents per gallon ; but 26 gallons leaked out, and he sold the remainder at 50 cents per gallon : did he gain or lose, and how much ? 33. Two persons counting their money, together they had 342 dollars ; but one had 28 dollars more than the other : how many had each ? 34. Mrs. Louisa Wilsie has 3 houses valued at 12530 dol- lars, 11324 dollars, and 9875 dollars : also a farm worth 6720 dollars. She had a daughter and 2 sons. To the daughter she gives one-third the value of the houses and one-fourth the value of the farm, and then divides the remainder equally among the boys : how much did each receive ? 35. A person having a salary of 1500 dollars, saves at the end of the year 405 dollars : what were his average daily expenses, allowing 365 days to the year ? 36. Mr. Bailey has 7 calves worth 4 dollars apiece, 9 sheep worth 3 dollars apiece, and a fine horse worth 175 dollars. He exchanges them for a yoke of oxen worth 125 dollars and a colt worth 65 dollars, and takes the balance in hogs at 8 dollars apiece : how many does he take ? 37. Mr. Snooks, the tailor, bought of Mr. Squire, the mer- chant, 4 pieces of cloth ; the first and second pieces each measured 45 yards, the third 47 yards, and the fourth 53 yards ; for the whole he paid 760 dollars : what did he pay for 35 yards ? 38. Mr. Jones has a farm of 250 acres, worth 125 dollars per acre, and offers to exchange with Mr. Gushing, whose farm contains 185 acres, provided Mr. Gushing will pay him 20150 dollars difference: what was Mr. Cushing's farm valued at per acre ? 78 APPLICATIONS. 39. The volcano in the island of Bourbon, in 1796, threw out 45000000 cubic feet of lava : how long would it take 25 carts to carry it off, if each cart carried 12 loads a day, and 40 cubic feet at each load ? 40. The income of the Bishop of Durham, in England, is 292 dollars a day ; how many clergymen would this support in a salary of 730 dollars per annum ? 41. The diameter of the earth is 7912 miles, and the diame- ter of the sun 112 times as great : what is the diameter of the sun? 42. By the census of 1850, the whole population of the United States was 23191876 ; the number of births for the previous year was 629444 and the number of deaths 324394 : supposing the births to be the only source of increase, what was the population at the beginning of the previous year ? 43. Mr. Sparks bought a third part of neighbor Spend- thrift's farm for 2750 dollars. Mr. Spendthrift then sold half the remainder at an advance of 250 dollars, and then Mr. Sparks bought what was left at a further advance of 250 dollars : how much money did Mr. Sparks pay Mr. Spend- thrift, and what did he get for his whole farm ? 44. George Wilson bought 24 barrels of pork at 14 dollars a barrel ; one-fourth of it proved damaged, and he sold it at half price, and the remainder he sold at an advance of 3 dol- lars a barrel : did he make or lose by the operation, and how much ? 45. A miller bought 320 bushels of wheat for 576 dollars, and sold 256 bushels for 480 dollars : what did the remain- der cost him per bushel ? 46. A merchant bought 117 yards of cloth for 702 dollars, and sold 76 yards of it at the same price for which he bought it ; what did the cloth sold amount to ? 47. If 46 acres of land produce 2484 bushels of corn ; how many bushels will 1 20 acres produce ? 48. Mr. J. Williams goes into business with a capital of 25000 dollars ; in the first year he gains 2000 ; in the second year 3500 dollars ; in the third year 4000 dollars ; he then invests the whole in a cargo of tea and doubles his money ; he then took out his original capital and divided the residue equally among his 5 "children : what was the portion of each ? UNITED STATES MONEY. 79 UNITED STATES MONEY. 81. Numbers are collections of units of the same kind. In forming these collections, we first collect the lowest or pri- mary units, until we reach a certain number ; we then change the unit and make a second collection, and after reaching a certain number we again change the unit, and so on. In abstract numbers, we first collect the units 1 till we reach ten ; we then change the unit, to 1 ten, and collect till we reach 10 ; we then change the unit to 100, and so on. A SCALE expresses the relations between the orders of units, in any number. There are two kinds of scales, uniform and varying. In the abstract numbers, the scale is uniform, the units of the scale being 10, at every step. 82. United States money is the currency established by Con- gress, A.D. 1786. The names or denominations of its units are, Double Eagles, Eagles, Dollars, Dimes, Cents, and Mills. The coins of the United States are of gold, silver, and cop- per, and are of the following denominations : 1. Gold : Double-eagle, eagle, half-eagle, three-dollars, quarter-eagle, dollar. 2. Silver: Dollar, half-dollar, quarter-dollar, dime, half- dime, and three-cent piece. 3. Copper : Cent, half-cent. TABLE. 10 Mills make 1 Cent, Marked ct. 10 Cents - - 1 Dime, - - d. 10 Dimes - - 1 Dollar, - - $. 10 Dollars - - 1 Eagle, - - E. Mills. Cents. Dimes. Dollars. Eagles. 10 = 1 100 = 10 = 1 1000 = 100 = 10 = 1 10000 = 1000 = 100 = 10 = 1 81. "What are numbers? How are numbers formed? How are sim- ple numbers formed ? What is the scale ? What is the primary unit in simple numbers ? 80 UNITED STATES MONEY. 83. It is seen, from the above table, that in United States money, the primary unit is 1 mill ; the units of the scale, in passing from mills to cents, are 10. The second unit is 1 cent, and the units of the scale, in passing to dimes, are 10. The third unit is 1 dime, and the units of the scale in passing to dollars, are 10. The fourth unit is 1 dollar, and the units of the scale in passing to eagles, are 10. This scale is the same as in simple numbers ; therefore, The units of United States money may be added, sub- tracted, multiplied, and divided, by the same rules that have already been given for simple numbers. NUMERATION TABLE. 5 7, is read 5 cents and 7 mills, or 57 mills. 1 6 4, - - 16 cents and 4 mills, or 164 mills. 6 2. 1 2 0, - - 62 dollars 12 cents and no mills. 27.623,- - 27 dollars 62 cents and 3 mills. 4 0. 4 1, - - 40 dollars 4 cents and 1 mill. The period, or separatrix, is generally used to separate the cents from the dollars. Thus $67.256 is read 67 dollars 25 cents and 6 mills. Cents occupy the two first places on the right of the period, and mills the third. United States money is read in dollars, cents and mills. 82. What is United States money? What are the names of its units ? What are the coins of the United States ? Which gold ? Which silver ? Which copper ? 83. In United States money what is the primary unit? What is the Hcale in passing from one denomination to another? I low does this compare with the scale in simple numbers ? What then follows V What is used to separate dollars from cents ? How is United States money read ? 84. What is reduction ? How many kinds of reduction are there ? Name them. How may cents be changed into mills? How may dol- lars be changed into cents ? How into mills ? UNITED STATES MONEY. 81 REDUCTION OF UNITED STATES MONEY. 84. Reduction of United States Money is changing the unit from one denomination to that of another, without altering the value of the number. It is divided into two parts : 1st. To reduce from a greater unit to a less, as from dol- lars to cents. 2d. To reduce from a less unit to a greater, as from mills to dollars. 85. To reduce from a greater unit to a less. From the table it appears, 1st. That cents may be changed into mills by annexing one cipher. 2d. That dollars may be changed into cents by annexing two ciphers, and into mills by annexing three ciphers. 3d. That eagles may be changed into dollars by annexing one cipher. The reason of these rules is evident, since 10 mills make a cent, 100 cents a dollar, and 1000 mills a dollar and 10 dollars 1 eagle. EXAMPLES. 1. Reduce 25 eagles, 14 dollars, 85 cents and 6 mills to the denomination of mills. OPERATION. 25 eagles =250 dollars, add 14 dollars, "264 dollars =2 64 00 cents, add - 85 cents, 26485 cents=264850 mills, add - - 6 mills, Ans. 264856 mills. 2. In 3 dollars 60 cents and 5 mills, how many mills ? 3 dollars =300 cents, 60 cents, 160 = 3600 mills, to which add the 5 mills. 6 82 REDUCTION OF 3. In 37 dollars 31 cents 8 mills, how many mills ? 4. In 375 dollars 99 cents 9 mills, how many mills ? 5. How many mills in 67 cents ? 6. How many mills in $54 ? 7. How many cents in $125 ? 8. In $400, how many cents ? How many mills ? 9. In $375, how many cents ? How many mills ? 10. How many mills in $4 ? In $6 ? In $10.14 cents. 11. How many mills in $40.36 cents 8 mills ? 12. How many mills in $71.45 cents 3 mills ? 86. To reduce from a less unit to a greater. 1. How many dollars, cents and mills in 26417 mills? ANALYSIS. We first divide the mills by 10, OPERATION. giving 2641 cents and 7 mills over; we then 10)264117 divide the cents by 100, giving 26 dollars, and 100)26141 41 cents over : hence the answer is 26 dollars *!> . -. *, 41 cents and 7 mills : therefore, I. To reduce mills to cents : cut off the right hand figure. II. To reduce cents to dollars : cut off the two right hand figures: and, III. To reduce mills to dollars : cut off the three right hand figures. EXAMPLES. 1. How many dollars cents and mills are there in 67897 mills ? 2. Set down 104 dollars 69 cents and 8 mills. 3. Set down 4096 dollars 4 cents and 2 mills. 4. Set down 100 dollars 1 cent and 1 mill. 5. Write down 4 dollars and 6 mills. 6. Write down 109 dollars and 1 mill. 7. Write down 65 cents and 2 mills. 8. Write down 2 mills. 9. Reduce 1607 mills, to dollars cents and mills. 10. Reduce 170464 mills, to dollars cents and mills. IK Reduce 8674416 mills, to dollars cents and mills. 12. Reduce 94780900 mills, to dollars cents and mills. 13. Reduce 74164210 mills, to dollars cents and mills. 8G. How do you change mills into cents ? How do you change cento Into dollars ? How do you change mills to dollars ? UNITED STATES MONEY. 83 87. One number is said to be an aliquot part of another, when it is contained in that other an exact number of times. Thus ; 50 cents, 25 cents, &c., are aliquot parts of a dollar : so also 2 months, 3 months,. 4 months and 6 months are ali- quot parts of a year. The parts of a dollar are sometimes expressed fractionally, as in the following TABLE OF ALIQUOT PARTS. $1 =100 cents. | of a dollar = 50 cents. | of a dollar = 33 J cents. J of a dollar = 25 cents, of a dollar = 20 cents. I of a dollar^ 121 cents. fa of a dollar = 10 cents. ^ of a dollar = 6J cents, z^j- of a dollar = 5 cents, of a cent = 5 mills. ADDITION OF UNITED STATES MONEY. 1. Charles gives 9| cents for a top, and 3J cents for 6 quills : how much do they all cost him ? 2. John gives $1.37 for a pair of shoes, 25 cents for a penknife, and 12 J cents for a pencil : how much does he pay for all ? OPERATION. ANALYSIS. We observe that half a cent is equal $1.375 to 5 mills. We then place the mills, cents and dol- '25 lars in separate columns. We then add as in simple I9f\ numbers. i - J $1.750 OPERATION. 3. James gives 50 cents for a dozen oranges, $0.50 12| cents for a dozen apples: and 30 cents for .125 a pound of raisins : how much for all ? .30 $0.925 ' 88. Hence, for the addition of United States money, we have the following RULE. I. Set down the numbers so that units of the same value shall stand in the same column. 87. What is an aliquot part ? How many cents in a dollar ? In half a dollar ? In a third of a dollar ? In a fourth of a dollar ? 84 APPLICATIONS IN II. Add up the several columns as in simple numbers, and place the separating point in the sum directly under that in the columns. PROOF. The same as in simple numbers. EXAMPLES. 1. Add $61.214. $10.049, $6.041, $0.271, together. (1.) (2.) (3.) $ cts. m. $ cts. m. $ cts. m. 67.214 59.316 81.053 10.049 87.425 67.412 6.041 48.872 95.376 0.271 56.708 87.064 $83.575 $330.905 APPLICATIONS. 1. A grocer purchased a box of candles for 6 dollars 89 cents : a box of cheese for 25 dollars 4 cents and 3 mills ; a keg of raisins for 1 dollar 12| cents, (or 12 cents and 5 mills ;) and a cask of wine for 40 dollars 37 cents 8 mills : what did the whole cost him ? 2. A farmer purchased a cow for which he paid 30 dollars and 4 mills ; a horse for which he paid 104 dollars 60 cents and 1 mill ; a wagon for which he paid 85 dollars and 9 mills : how much did the whole cost ? 3. Mr. Jones sold farmer Sykes 6 chests of tea for $75.641 ; 9 yards of broadcloth for $27.41 ; a plow for $9.75 ; and a harness for $19.674 : what was the amount of the bill ? 4. A grocer sold Mrs. Williams 18 hams for $26.497 ; a bag of coffee for $17.419 ; a chest of tea for $27.047 ; and a firkin of butter for $28.147 : what was the amount of her bill? 5. A father bought a suit of clothes for each of his four boys ; the suit of the eldest cost $15.167 ; of the second, $13.407 ; of the third, 12.75 ; and of the youngest, $11.047 : how much did he pay in all ? 88. How do you set down the numbers for addition ? How do you add up the columns ? How do you place the separating point ? How do you prove addition ? UNITED STATES MONEY. 85 6. A father has six children ; to the first two he gives each $375.416 ; to each of the second two, $287.55 ; to each of the remaining two, $259.004 : how much did he give to them all? 7. A man is indebted to A, $630.49 ; to B, $25 ; to C, 87 J cents ; to D, 4 mills : how much does he owe ? 8. Bought 1 gallon of molasses at 28 cents per gallon ; a half pound of tea for 78 cents ; a piece of flannel for 12 dol- lars 6 cents and 3 mills ; a plow for 8 dollars 1 cent and 1 mill ; and a pair of shoes for 1 dollar and 20 cents : what did the whole cost ? 9. Bought 6 pounds of coffee for 1 dollar 12J cents ; a wash-tub for 75 cents 6 mills ; a tray for 26 cents 9 mills ; a broom for 27 cents ; a box of soap for 2 dollars 65 cents 7 mills ; a cheese for 2 dollars 87^ cents : what is the whole amount ? 10. What is the entire cost of the following articles, viz. : 2 gallons of molasses, 57 cents ; half a pound of tea, 37| cents ; 2 yards of broadcloth, $3.37| cents ; 8 yards of flan- nel, $9.875 ; two skeins of silk, 12| cents, and 4 sticks of twist, 8i cents ? SUBTRACTION OF UNITED STATES MONEY. 1. John gives 9 cents for a pencil, and 5' cents for a top, how much more does he give for the pencil than for the top ? 2. A man buys a cow for $26.37, and a calf for $4.50 : how much more does he pay for the cow than for the calf ? OPERATION. NOTE. We set down the numbers as in addition, $26.37 and then subtract them as in simple numbers. 4 50 $21.87 89. Hence, for subtraction of United States money, we have the following RULE. I. Write the less number under the greater so thai units of the same value shall stand in the same column. 89. How do you set down the numbers for subtraction ? How do you subtract them ? Where do you place the separating point in the remainder ? How dc you prove subtraction ? 86 SUBTRACTION OF II. Subtract as in simple numbers, and place the separating point in the remainder directly under that in the columns. PROOF. The same as in simple numbers. EXAMPLES. (I-) (2.) From $204.679 From $8976.400 Take 98.714 Take 610.098 Remainder $105.965 Remainder $8366.302 (3.) (4.) (5.) $620.000 $327.001 $2349 19.021 2.090 29.33 $600.979 $324.911 $2319.67 6. What is the difference between $6 and 1 mill ? Between $9.75 and 8 mills ? Between 75 cents and 6 mills? Between $87.354 and 9 mills? 7. From $107.003 take $0.479. 8. From $875.043 take $704.987. 9. From $904.273 take $859.896. APPLICATIONS. 1. A man's income is $3000 a year ; he spends $187.50 : how much does he lay up ? 2. A man purchased a yoke of oxen for $78, and a cow for $26.003 : how much more did he pay for the oxen than for the cow ? 3. A man buys a horse for $97.50, and gives a hundred dollar bill : how much ought he to receive back ? 4. How much must be added to $60.039 to make the sum $1005.40? 5. A man sold his house for $3005, this sum being $98.039 more than he gave for it : what did it cost him ? 6. A man bought a pair of oxen for $100, and sold th'em again for $7 5.37 J : did he make or lose by the bargain, and how much ? 7. A man starts on a journey with $100 ; he spends $87.57 : how much has he left? 8. How much must you add to $40.173 to make $100? UNITED STATES MONEY. 87 9. A man purchased a pair of horses for $450, but finding one of them injured, the seller agreed to deduct $106.325 : what had he to pay ? 10. A farmer had a horse worth $147.49, and traded him for a colt worth but $35.048 : how much should he receive in money ? 11. My house is worth $8975.034; my barn $695.879: what is the difference of their values ? 12. What is the difference between nine hundred and sixty- nine dollars eighty cents and 1 mill, and thirty-six dollars ninety-nine cents and 9 mills ? MULTIPLICATION OF UNITED STATES MONEY. 1. John gives 3 cents apiece for 6 oranges : how much do they cost him ? 2. John buys 6 pairs of stockings, for which he pays 25 cents a pair : how much do they cost him ? 3. A farmer sells 8 sheep for $1.25 each : how much does he receive for them ? OPERATION. ANALYSIS. We multiply the costs of one sheep by $1.25 the number of sheep, and the product is the entire ' o cost. $10.00 90. Hence, for the multiplication of United States money by an abstract number, we have the following RULE. I. Write the money for the multiplicand, and the abstract number for the multiplier. II. Multiply as in simple numbers, and the product will be the answer in the lowest denomination of the multi- plicand. III. Reduce the product to dollars, cents and mills. PROOF. Same as in simple numbers EXAMPLES. 1. Multiply 385 dollars 28 cents and 2 mills, by 8. OPERATION. (2.) $385.282 $475.87 8 9 Product $3082.256 Product $4282.83 88 MULTIPLICATION OF 3. What will 55 yards of cloth come to at 37 cents per yard? 4. What will 300 bushels of wheat come to at $1.25 per bushel ? 5. What will 85 pounds of tea come to at 1 dollar 37 cents per pound ? 6. What will a firkin of butter containing 90 pounds come to at 25J cents per pound ? 7. What is the cost of a cask of wine containing 29 gal- lons, at 2 dollars and 75 cents per gallon ? 8. A bale of cloth contains 95 pieces, costing 40 dollars 37 J cents each : what is the cost of the whole bale ? 9. What is the cost of 300 hats at 3 dollars and 25 cents apiece ? 10. What is the cost of 9704 oranges at 3J cents apiece ? OPERATION. NOTE. We know that the product of two num- bers contains the same number of units, whichever be used as the multiplier (Art. 48). Hence, we may multiply 9704 by 3^ if we assign the proper unit (1 cent) to the product. $339.64 11. What will be the cost of 356 sheep at 3J dollars a head ? 12. What will be the cost of 47 barrels of apples at 1 j dollars per barrel ? 13. What is the cost of a box of oranges containing 450, at 2 cents apiece ? 14. What is the cost of 307 yards at linen of 68J cents per yard ? 15. What will be the cost of 65 bushels of oats at 33* cents a bushel ? ANALYSIS. If the price were 1 dollar a bushel, OPERATION. the cost would be as many dollars as there are 3)65.000 bushels. But the cost is 38^ cents = of a dollar : .. flrpa hence, the cost will be as many dollars as 3 is con- tained times in 65=21 dollars, and 2 dollars over, which is re- 90. How do you multiply United States money ? What will be the denomination of the product ? How will you then reduce it to dollars and cents ? How do you prove multiplication ? UNITED STATES MONEY. 89 duced to cents by annexing two ciphers, and to mills by annexing three ; then, dividing the cents and mills by 3, we have the entire cost: hence, 91. To find the cost, when the price is an aliquot part of a dollar. Take such a part of the number which denotes the commo- dity, as the price is of I dollar. EXAMPLES. 1. What would be the cost of 345 pounds of tea at 50 cents a pound ? 2. What would 675 bushels of apples cost at 25 cents a bushel ? 3. If 1 pound of butter cost 12| cents, what will 4 firkins cost, each weighing 56 pounds ? 4. At 20 cents a yard, what will 42 yards of cloth cost ? 5. At 33 J cents a gallon, what will 136 gallons of mo- lasses cost ? OPERATION. 6. What will 1276 yds. 4)$1276 cost at 1 dollar a yard, of cloth cost at $1.25 a 319 cost at 25 cts. a yard, yard ? $1595 C ost at $1.25 a yard. 7. What would be the cost of 318 hats at $1.12J apiece ? 8. What will 2479 bushels of wheat come to at $1.50 a bushel ? 9. At $1.33J a foot, what will it cost to dig a well 78 feet deep ? 10. What will be the cost of 936 feet of lumber at 3 dollars a hundred ? ANALYSIS. At 3 dollars a foot the cost would be OPERATION. 936x3=2808 dollars ; but as 3 dollars is the price 935 of 100 feet, it follows that 2808 dollars is 100 times the cost of the lumber: therefore, if we divide 2808 dollars by 100 (which we do by cutting off two $28.08 of the right hand figures (Art. 73), we shall obtain the cost. NOTE. Had the price been so much per thousand, we should have divided by 1000, or cut off three of the right hand figures : hence, 91. How do you find the cost of several things when the price is an aliquot part of a dollar ? 90 MULTIPLICATION OF 92. To find the cost of articles sold by the 100 or 1000 ; Multiply the quantity by the price ; and if the price be by the 100, cut off two figures on the right hand of the product ; if by the 1000, cut off three, and the remaining figures will be the answer in the same denomination as the price, which if cents or mills, may be reduced to dollars. EXAMPLES. 1. What will 4280 bricks cost at $5 per 1000 ? 2. What will 2673 feet of timber cost at $2.25 per 100 ? 3. What will be the cost of 576 feet of boards at $10.62 per 1000 ? 4. What is the value of 1200 feet of lathing at 7 dollars per 1000 ? 5. David Trusty, Bought of Peter Bigtree. 2462 feet of boards at $7. per 1000. 4520 u ' 9.50 600 " scantling 1 11.37 960 " timber 1 15. 1464 " lathing .75 per 100. 1012 " plank ' 1.25 Received Payment, Peter Bigtree, 6. What is the cost of 1684 pounds of hay at $10.50 per ton? ANALYSIS. Since there are OPERATION. 2000*. in a ton, the cost of 2)10.50 ?o r 00 " ^$5^ ~55 price of 1000ft S . cents. Multiply this by the 1684 number of pounds (1684), and $g 841QO Ans. cut off three places from the right,, in addition to the two places before cut off for cents : hence, 93. To find the cost of articles sold by the ton : Multiply one-half the price of a ton by the number of pounds' and cut off three figures from the right hand of the product. The remaining figures will be the answer i the same denomination as the price of a ton. 92. How do you find the cost of articles sold by the 100 or 1000 ? UNITED STATES MONEY. 91 EXAMPLES. 1. What will 3426 pounds of plaster cost at $3.48 per ton? 2. What will be the cost of the transportation of 6742 pounds of iron from Buffalo 'to New York, at $7 per ton ? 3. What will be the cost of 840 pounds of hay at $9.50 per ton? at $12? at $15.84 ? at $10.36 ? at $18.75? DIVISION OP UNITED STATES MONEY. 94. To divide a number expressed in dollars, cents or mills, into any number of equal parts. RULE. I. Reduce the dividend to cents or mills, if necessary. II. Divide as in simple numbers, and the quotient will be the answer in the lowest denomination of the dividend : this may be reduced to dollars, cents, and mills. PROOF. Same as in division of simple numbers. NOTE. The sign + is annexed in the examples, to show that there is a remainder, and that the division may be continued. EXAMPLES. 1. Divide $4.624 by 4 : also, $87.256 by 5. OPERATION. OPERATION. 4)$4.624 5j$87.256 $1.156 $17.454 2. Divide $37 by 8. ANALYSIS. In this example we first reduce the OPERATION. $37 to mills by annexing three ciphers. The quo- 8)$37,000 tient will then be mills, and can be reduced to dol- <fe //fio^ lars and cents, as before. v 4,bJo 3. Divide $56.16 by 16. 4. Divide $495.704 by 129. 5. Divide $12 into 200 equal parts. 6. Divide $400 into 600 equal parts. 7. Divide $857 into 51 equal parts. 8. Divide $6578.95 into 157 equal parts. 93. How do you find the cost of articles sold by the ton ? 94. What is the rule for division of United States money ? How do you prove division ? How do you indicate that the division may be continued ? 92 DIVISION OF 95. The quantity, and the cost of a quantity given, to find the price of one thing (Art. 80). Divide the cost by the quantity. 9. Bought 9 pounds of tea for $5.85 ; what was the price per pound ? 10. Paid $29.68 for 14 barrels of apples: what was the price per barrel ? 11. If 27 bushels of potatoes cost $10.125, what is the price of a bushel ? 12. If a man receive $29.25 for a month's work, how much is that a day, allowing 26 working days to the month ? 13. A produce dealer bought 3 barrels of eggs, each con- taining 150 dozens, for which he paid $63 : how much did he pay a dozen ? 14. A man bought a piece of cloth containing 72 yards, for which he paid $252 : what did he pay per yard ? 15. If $600 be equally divided among 26 persons, what will be each one's share ? 16. Divide $18000 into 40 equal parts: what is the value of each part ? 17. Divide $3769.25 into 50 equal parts: what is one part? 18. A farmer purchased a farm containing 725 acres, for which he paid $18306.25 : what did it cost him per acre ? 19. A merchant buys 15 bales of goods at auction, for which he pays $1000 : what do they cost him per bale ? 20. A drover pays $1250 for 500 sheep ; what shall he sell them for apiece, that he may neither make nor lose by the bargain ? 21. The dairy of a farmer produces $600, and he has 25 cows : how much does he make by each cow ? 22. A farmer receives $840 for the wool of 1400 sheep : how much does each sheep produce him ? 23. A merchant buys a piece of goods containing 105 yards, for which he pays $262.50 ; he wishes to sell it so as to make $52.50 : how much must he ask per yard? 90. When the price of one and the cost of a quantity are given, to find the quantity (Art. 80). NoraThe divisor and dividend must both be reduced to the lowest unit named in either before dividing. UNITED STATES MONEY. 93 Divide the cost by the price. 24. If I pay $4.50 a ton for coal, how much can I buy for $67.50 ? 25. At $7 a barrel, how much flour can be bought for $178.50? 26 How many pounds of tea can be bought for $6.75, at 75 cents a pound ? 27. What number of barrels of apples can be bought for $47.50, at $2.37 J a barrel? 28. At 44 cents a bushel, how many bushels of oats can be bought for $14.30 ? 29. At 34 cents a bushel, how many barrels of apples can I buy for $13.60, allowing 2J bushels to the barrel? 30. If 1 acre of land costs $28.75, how much can be bought for $3220 ? 31. Paid $40.50 for a pile of wood, at the rate of $3.37J a cord, how much was there in the pile ? 32. How many sheep can be bought for $132, at $1.37| a head ? 33. At $4.25 a yard, how many yards of cloth can be bought for $68 ? 34. At $1.12J a day, how long would it take a person to earn $157.50. APPLICATIONS IN THE FOUR PRECEDING RULES. NOTE. See and repeat Rule page 53 : also the three rules page 74. 1. If 1 yard of cloth costs 3 J dollars, what will 8 yards cost ? 2. If 1 ton of hay costs $14 J, what will 9 tons cost ? 3. If 1 calf costs $4 J, what will 12 calves cost ? 4. Mr. Jones bought 250 bushels of oats, for which he paid $156.25 : how much did they cost him a bushel ? 5. If 12 tons of hay cost 150 dollars, what does 1 ton cost ? 8 tons ? 50 tons ? 6. If 9 dozen of spelling books cost $7.875, what will 1 dozen cost ? 6 dozen ? 8 dozen ? 7. If 75 bushels of wheat cost $131.25, how much will 1 bushel cost ? 8 bushels ? 120 bushels ? 8. If 320 pounds of coffee cost $44.80 cents, how much will 1 pound cost ? What will 575 pounds cost ? 94: APPLICATIONS IN 9. Mr. James B. Smith bought 9 barrels of sugar, each weighing 216 pounds, for which he paid $116.64 : how much did he pay a pound ? 10. If 40 tons of hay cost $580, how much is that per ton ? What would 70 tons cost at the same rate ? 11. If Mr. Wilson has $120 to buy his winter wood, and wood is $4 a cord, how many cords can he buy ? 12. At 6 dollars a yard, how many yards of cloth can be bought for $24 ? How many for $36 ? 13. A farmer sold a yoke of oxen for $80.75 ; 6 cows for $29 each ; 30 sheep at $2.50 a head ; and 3 colts, one for $25, the other two for $30 apiece ; what did he receive for the whole lot ? 14. A merchant buys 6 bales of goods, each containing 20 pieces of broadcloth, and each piece of broadcloth contained 29 yards ; the whole cost him $15660 ; how many yards of cloth did he purchase, and how much did it cost him per yard? 15. A person sells 3 cows at $25 each ; and a yoke of oxen for $65 ; he agrees to take in payment 60 sheep : how much do his sheep cost him per head ? 16. A man dies leaving an estate of $33000 to be equally divided among his 4 children, after his wife shall have taken her third. What was the wife's portion, and what the part of each child ? 17. A person settling with his butcher, finds that he is charged with 126 pounds of beef at 9 cents per pound ; 85 pounds of veal at 6 cents per pound ; 6 pairs of fowls at 37 cents a pair ; and three hams at $1,50 each : how much does he owe him ? 18. A farmer agrees to furnish a merchant 40 bushels of rye at 62 cents per bushel, and to take his pay in coffee at 16 cents per pound : how much coffee will he receive ? 19. A farmer has 6 ten-acre lots, in each of which he pas- tures 6 cows ; each cow produces 112 pounds of butter, for 1 which he receives 18 \ cents per pound ; the expenses of each cow are 5 dollars and a half : how much does he make by his dairy ? 20. Bought a farm of W. N. Smith for 2345 dollars, a span of horses for 375 dollars, 6 cows at 36 dollars each ? I paid him 520 dollars in cash, and a village lot worth 1500 dollars : how many dollars remain unpaid ? UNITED STATES MONEY. 95 BILLS OF PARCELS. (21.) New York, May 1st, 1854. Mr. James Spendthrift, Bought of Benj. SavedLl. 16 pounds of tea at 85 cents per pound - - - 27 pounds of coffee at 15J cents per pound - - 15 yards of linen at 66 cents per yard - - - - Received payment, Benj. Saveall. (22.) Albany, June 2d, 1854; Mr. Jacob Johns, Bought of Gideon Gould. 36 pounds of sugar at 9 J cents per pound - - 3 hogsheads of molasses, 63 galls, each, at 27 cents a gallon 5 casks of rice, 285 pounds each, at 5 cents per pound 2 chests of tea, 86 pounds each, at 96 cents per ) pound f Total cost, $ Received payment, For Gideon Gould, Charles Clark. <J23.) Hartford, November 21st, 1854. Gideon Jones, Bought of Jacob Thrifty. 69 chests of tea at $55.65 per chest - - - - 126 bags of coffee, 100 pounds each, at 12J ) cents per pound } 167 boxes of raisins at $2.75 per box - - - 800 bags of almonds at $18.50 per bag - - - 9004 barrels of shad at $7.50 per barrel - - - 60 barrels of oil, 32 gallons each, at $1.08 ) per gallon ) Amount, $ Received the above in full. Jacob Thrifty. 90 DENOMINATE NUMBERS. DENOMINATE NUMBERS. 97. A SIMPLE NUMBER is a unit or a collection of units. The unit may be either abstract or denominate. 98. A DENOMINATE NUMBER is a denominate unit or a collection of units : thus, 3 yards is a denominate number, in which the unit is 1 yard. 99. Numbers which have the same unit, are of the same denomination: and numbers having different units, are of different denominations. If two or more denominate num- bers, having different units, are connected together, forming a single number, such is called a compound denominate number. 100. There are eight different units in Arithmetic : 1st. The abstract unit : 2d. The unit of currency : 3d. The unit of length : 4th. The unit of surface : 5th. The cubic unit or unit of volume : 6th. The unit of weight : 7th. The unit of time : 8th. The unit of circular measure. ENGLISH MONEY. 101. The units or denominations of English money are guineas, pounds, shillings, pence, and farthings. TABLE. 4 farthings marked far make 1 penny, marked d. 12 pence - 1 shilling, s. 20 shillings - 1 pound, or sovereign, , 21 shillings - - . 1 guinea. far. d. s. 4 =1 48 =12 = 1 960 =240 =20 =1 NOTES. 1. The primary unit in English money is 1 farthing. The number of units in the scale, in passing from farthings to 97. What is a simple number ? 98. What is a denominate number ? 99. When are numbers of the same denomination ? When of differ- ent denominations ? If several numbers having different units are con- nected together, what is the number called ? 100. How many units are there in Arithmetic ? Name them, DENOMINATE NUMBERS. 97 pence, is 4 ; in passing from pence to shillings, 12 ; in passing from shillings to pounds, 20. 2. Farthings are generally expressed in fractions of a penny. Thus, 1 far.=tf.; 2 far.=\d. ; 3 far.=$d. 3. By reading the second table from right to left, we can see the value of any unit expressed in each of the lower denomina- tions. Thus, ld. = 4far.; 1*.= 12d.=4Stfar. ; l=20.= 240d. REDUCTION OF DENOMINATE NUMBERS. 102. Reduction is changing the unit of a number, without altering its value. 1. How many pence are there in 2s. &d. ? ANALYSIS. Since there are 12 pence in 1 shilling, there are twice 12, or 24 pence in 2 shillings : add the 6 pence : therefore, in 2s. 6d. there are 30 pence. 2. How many pence in 4 shillings? In 4s. Sd. ? In 5s. Sd. ? In 3s. Sd. ? In 6s. Id. ? 3. How many shillings in ,2 ? In 3 8s., how many ? 4. How many pence in 1 ? How many shillings in 2 8s. ? How many in ^3 7s. ? 5. How many shillings are there in 48 pence ? ANALYSIS. Since there are 12 pence in 1 shilling, there are as many shillings in 48 pence, as 12 is contained times in 48, which is 4: therefore, there are 4 shillings in 48 pence. 6. How many pounds in 40 shillings ? In 60 ? In 80 ? 103. From the above analyses we see, that reduction of denominate numbers is divided into two parts : 1st. To change the unit of a number from a higher deno- mination to a" lower. 2d. To change the unit of a number from a lower denomi- nation to a higher. 101. What are the denominations of English money ? Notes. 1 What is the primary unit in English money ? Name the units of the scale. 2. How are farthings generally expressed ? 3. How is the second table read ? What does it show ? 102. What is Reduction ? 103. Into how many parts is reduction divided ? What are tliey ? 7 98 REDUCTION OF PRINCIPLES AND EXAMPLES. 104. To reduce from a higher to a lower unit. 1. Reduce JE21 6s. Sd. to the denomination of farthings OPERATION. ANALYSIS. Since there are 20 shillings in 27 6s &d 2far 1, in 27 there are 27 times 20 shillings, ' on ' or 540 shillings, and 6 shillings added, make 546*. Since 12 pence make 1 shilling, we next multiply by 12, and then add Sd. to the product, giving 6560 pence. Since 4 far- pf . Rn , things make 1 penny, we next multiply by 4, and add 2 farthings to the product, giv- ing 26242 farthings for. the answer. 26242 NOTE. The units of the scale, in passing from pounds to shil- lings, are 20 ; in passing from shillings to pence they are 12 ; and in passing from pence to farthings, 4. Hence, to reduce from a higher to a lower unit, -we have the following RULE. Multiply the highest denomination by the units of the scale which connect it with the next lower, and add to the product the units of that denomination ; proceed in the same manner through all thd denominations, till the unit is brought to the required denomination. 105. To reduce from a lower unit to a higher. 1. Reduce 3138 farthings to pounds. OPERATION. ANALYSIS. Since 4 farthings 4)3138 make a penny, we first divide by 4. 1 0N ^ Q _ Since 12 pence make a shilling, we _ ' 2 J ar - rCTn - next divide by 12. Since 20 shil- 210)615 - - 4d. rem. lings make a pound, we next divide c " r ' by 20, and find that l38/ar.=3 - " 5s. 4d. 2 far. Ans. 3 5s. 4d. 2 far. Hence, to reduce from a lower to a higher denomination, we have the following RULE. I. Divide the given number by the units of the scale 104. How do you reduce from a higher to a lower unit? 105. How do you reduce from a lower to a higher unit? What will be" the unit of any remainder ? How do you prove reduction ? DENOMINATE NUMBERS. 99 which connect it with the next higher denomination, and set down the remainder, if there be one. II. Divide the quotient thus obtained by the units of the scale which connect it with the next higher denomination, and set down the remainder. III. Proceed in the same way to the required denomination, and the last quotient, with the several remainders annexed, ivill be the answer. NOTE. Every remainder will be of the same denomination as its dividend. PROOF. After a number has been reduced from a higher denomination to a lower, by the first rule, let it be reduced back by the second ; and after a number has been reduced from a lower denomination to a higher, by the second rule, let it be reduced back by the first rule. If the work is right, the results will agree. EXAMPLES. 1. Reduce 15 7s. &d. to pence. OPERATION. PROOF. 15 7s. Gd. 12)3690 20 2|0)30|7 ... 6^. rem. 307 15 . . . 7s. rem. 12 3690 Ans. 15 7s. bd. 2. In 31 8s. 9<1 3 far., how many farthings? Also proof. 3. In 87 14s. 8^d., how many farthings ? Also proof. 4. In 407 19s. 11 %d., how many farthings? Also proof. 5. In 80 guineas, how many pounds ? 6. In 1549 far., how many pounds, shillings and pence? 7. In 6169 pence, how many pounds ? LINEAR MEASURE. 100. This measure is used to measure distances, lengths, breadths, heights and depths, &c. 106. For what is Linear Measure used ? What are its denominations ? Repeat the table. What is a fathom? What is a hand? What are the units of the scale ? 100 REDUCTION OF TABLE. 12 inches, in. make 3 feet 5J yards or 16 J feet - 40 rods - 8 furlongs or 320 rods 3 miles 69J statute miles (nearly) or 60 geographical miles, 360 degrees, ft. 1 foot, 1 yard, 1 rod, 1 furlong, - 1 mile, 1 league, 1 degree of) marked Af>.n. i a rd. fur. mi. L. n. 12 36 198 7920 =3 = 16 = 66 yd. =1 = 220 the equator a circum'nce of the earth. rd. 63360 = 5280 = 1760 = 1 = 40 = 320 fur. _ i _ t = 8 mi. NOTES. 1. A fathom is a length of six feet, and is generally Bed to measure the depth of water. 2. A hand is 4 inches, used to measure the height of horses. 3. The units of the scale, in passing from inches to feet, are 12 ; in passing from feet to yards, 3 ; from yards to rods, 5 ; from rods to furlongs, 40 ; and from furlongs to miles, 8. 1. How many inches in 5 feet ? In 10 feet ? In 16 feet ? 2. How many yards in 36 feet ? In 54 feet ? In 96 ? 3. How many feet in 144 inches ? In 96 inches ? In 48 ? 4. How many furlongs in 3 miles ? In 6 miles ? In 8 ? EXAMPLES. 1. How many inches in &rd. 4yd. 2ft. 9in. OPERATION. 6rdL 4yd. 2ft. 9in. _M 3 34 37 yards. 3 113 feet. 12 1365 inches. 2. In 1365 inches, how many rods ? OPERATION. 12)1365 3)113 feet 9m. 5|)37 yards 2ft 11)74 6rd. Ans. Qrd. 4yd. 2ft. 9m. DENOMINATE NUMBERS. 101 NOTE. When we reduce rods to yards, we multiply by the scale 5i ; that is, we take 6 rods 5 and one-half times. When we reduce yards to rods, we divide by 5i, which is done by reducing the dividend and divisor to halves : the remainder is 8 half-yards, equal to 4 yards. 3. In 59wi. *lfur. 38rY?., how many feet ? 4. In 115188 rods, how many miles? 5. In 719??u'. I6rd. 6yd., how many feet? (6. In 118, how many miles? 7. In 54 45mi. 7/ur. 20rd. yd. 2ft. Win., how many Inches ? 8. In 481401716 inches, how many degrees, &c. ? CLOTH MEASURE. 107. Cloth measure is used for measuring all kinds of cloth, ribbont;, and other things sold by the yard. TABLE. nail, marked na. quarter of a yard, qr. Ell Flemish, E. Fl. yard, - yd. Ell English, , E. E. 2J inches, in. 4 nails make 1 1 3 quarters - 4 quarters - 5 quarters - 1 1 1 in. na. 2J 1 qr. 9 =4 = 1 27 = 12 = 3 36 = 16 = 4 45 = 20 = 5 E.Fl = 1 yd. - l E. E. = 1 NOTE. The units in this measure are, inches, nails, quarters, Klls Flemish, yards, and Ells English. 1. In 9 inches, how many nails ? How many nails in 1 yard ? In 2 yards ? In 6 ? In 8 ? 2. In 4 yards, how many quarters ? How many quarters in 8 yards ? In 7 how many ? 3. How many quarters in 12 nails? In 16 nails? In 20 nails? In 36? In 40 ? 107. For what is cloth measure used ? What are its denominations ? Repeat the table. What are the units of this measure ? 102 REDUCTION OF 1. How many nails are there in 35yd. 3^r. 3na. ? OPEKATION. 35t/d. 3(? 4 EXAMPLES. 143 quarters. 4 575 nails. 2. In 575 nails, how many yards ? OPERATION. 4)575 4)143 3na. 35 3 jr. Ans. . 3gr. 3. In 49 E. E., how many nails ? 4. In 51 i?. FL, 2qr. 8na., how many nails ? 5. In 3278 nails, how many yards ? 6. In 340 nails, how many Ells Flemish ? 7. In 4311 inches, how many E. E. ? SQUARE MEASURE. 108. Square measure is used in measuring land, or anything in which length and breadth are both considered. 1 Foot. A square is a figure bounded by four equal lines at right angles to each other. Each line is called a side of the square. If each side be one foot, the figure is called a square foot. If the sides of the square be each one yard, the square is called a square yard. In the large square there are nine small squares, the sides of which are each one foot. Therefore, the square yard contains 9 square feet. The number of small squares that is contained in any large square is always equal to the product of two of the sides of the large square. As in the figure, 3 x3~9 square feet. The number of square inches contained in a square foot is equal to 12 x 12=144. 108. For what is Square Measure used? What is a square? If each side be one foot, what is it called ? If each side be a yard, whnt is it called ? How many square feet docs the square yard contain ? How is the number of small squares contained in a large square found ? Repeat the table. What are the units of the scale ? DENOMINATE NUMBERS. 103 TABLE. 144 square inches, sq. in., make 1 square foot, 9 square feet 30 J square yards - 40 square rods or perches - 4 roods - 640 acres - Sq.ft. I square yard, Sq. yd. 1 square rod or perch, P. 1 rood, - E. 1 acre, - A. 1 square mile, M. Sq. in. 144 1296 39204 1568160 6272640 _Sq.ft. = 9 = 272J = 10890 = 43560 Sq. yd. 1 301 1210 4840 P. 1 40 160 E. = 1 = 4 =1. NOTE. The uDits of the scale are 144, 9, 30L 40, 4 and 640. 1. How many square inches in 2 square feet? How many square feet in 3 square yards ? How many in 6 ? In 8 ? 2. How many perches in 1 rood ? In 3 roods ? How many roods in 4 acres ? In 8 ? In 12 ? 3. How many perches in an acre ? How many in 2 acres ? How many square yards in 81 square feet? SURVEYORS' MEASURE. 109. The Surveyor's or Gunter's chain is generally used in surveying land. It is 4 poles or 66 feet in length, and is divided into 100 links. inches make 4 .rods or 66/X 80 chains - 1 square chain 10 square chains TABLE. 1 link, marked - I. 1 chain, - c. I mile, - mi. 16 square rods or perches, P. 1 acre, - A. NOTE. 1. Land is generally estimated in square miles, acres> roods, and square rods or perches. 2. The units of the scale are 7 f 9 o 2 -, 4, 80. 109. What chain is used in land surveying ? What is its length ? How is it divided? Repeat the table. In what is land generally esti- mated ? What are the units of the scale ? 104 REDUCTION OF 1. How many rods in 1 chain ? How many in 4 ? In 5 ? 2. How many chains in 1 mile ? In 2 miles ? In 3 ? 3. How many perches in 1 square chain ? In 4 ? In 6 ? 4. How many square chains in 2 acres ? How many perches in 3 acres ? In5? In 6? EXAMPLES. 1. How many perches in 32Jf. 25A 35. 19P.? OPERATION. 323f. 25A 3P. 19P. 640 20505 acres. 4 82023 roods. 40 2. How many square miles, &c., in 3280989P.1 OPERATION. 40)3280939 19P. 4)82023 37?. 640)20505 25A 32 , Ans. 321T. 25 A ZR 19P. 3280939 perches. 3. In 19A 272. 37P., how many square rods ? 4. In 175 square chains, how many square feet ? 5. In 37456 square inches, how many square feet ? 6. In 14972 perches, how many acres ? 7. In 3674139 perches, how many square miles? 8. Mr. Wilson's farm contains 104A 3P. and 19P. ; he paid for it at the rate of 75 cents a perch : what did it cost? 9. The four walls of a room are each 25 feet in length and 9 feet in height and the ceiling is 25 feet square : how much will it cost to plaster it at 9 cents a square yard ? CUBIC MEASURE. 110. Cubic measure is used for measuring stone, timber, earth, and such other things as have the three dimensions, length, breadth, and thickness. TABLE. 1728 cubic inches, Cu. in., make 1 cubic foot, Cu. ft. 27 cubic feet, - 1 cubic yard, Cu. yd. 40 feet of round or ) -, . n, 50 feet of hewn timber, J 42 cubic feet, - 1 ton of shipping, T. 16 cubic feet, - - 1 cord foot, C.ft. 8 cord feet, or ) . , r 128 cubic feet, \ ' l cord ' DENOMINATE NUMBERS. 105 NOTE. 1. A cord of wood is a pile 4 feet wide, 4 feet high, and 8 feet long. 2. A cord foot is 1 foot in length of the pile which makes a cord. 3. A CUBE is a figure bounded by six equal squares, called faces; the sides of the squares are called edges. 4. A cubic foot is a cube, each of whose faces is a square foot, its edges are each 1 foot. 5. A cubic yard is a cube, each of whose edges is 1 yard. 6. The base of a cube is the face on which it stands If the edge of the cube is one yard, it will contain 3x3=9 square feet ; therefore, 9 ^ __ cubic feet can be placed on the base, j and hence, if the figure were 1 foot thick, it would contain 9 cubic feet ; d feet - 1 if it were 2 feet thick it would contain 2 tiers of cubes, or 18 cubic feet ; if it were 3 feet thick, it would contain 27 cubic feet ; hence, The contents of a figure of this form are found by multi- plying the length, breadth, and thickness together. 7. A ton of round timber, when square, is supposed to produce 40 cubic feet ; hence, one-fifth is lost by squaring. 1. In 1 cubic foot, how many cubic inches? How many in 2 ? In 3 ? 2. In 1 cubic yard, how many cubic feet ? How many in 2 ? In 4 ? In 6 ? 3. How many cord feet in 3 cords of wood ? In 5 ? In 6 ? 4. How many cubic feet in 2 cords ? In half a cord, how many ? How many in a quarter of a cord ? 5. How many cubic yards in 54 cubic feet ? In 81 ? 6. In 120 feet of round^ timber, how many tons ? 7. How many tons of shipping in 84 cubic feet ? In 168 ? 8. How many cords of wood in 64 cord feet ? In 96 ? In 128? 9. How many cubic feet in a stone 8 feet long, 3 feet wide and 2 feet thick ? 110. For what is cubic measure used ? What are its denominations ? What is a cord of wood ? What is a cord foot ? What is a cube ? What is a cubic foot ? What is a cubic yard ? How many cubic feet in a cubic yard? What are the contents of a solid equal to? Repeat the table. What are the units of the scale ? 106 REDUCTION OF EXAMPLES. 1. In 15cw. yd. IScu. ft. 16cw. in., how many cubic inches ? OPERATION. cu. yd. cu. ft. cu. in. 15 18 16 113 31 423x1728 + 16=730960. 2. In 730960 cubic inch- es, how many cubic yards, &c.? OPERATION. 1728)730960 cu. in. 27^423 cu. ft. 16 15ctt.yd.18 cu. yd. cu.ft. cu. in. Ans. 15 18 16 3. How many small blocks 1 inch on each edge can be sawed out of a cube 7 feet on each edge, allowing no waste for sawing ? 4. In 25 cords of wood, how many cord feet ? How many cubic feet ? 5. How many cords of wood in a pile 28 feet long, 4 feet wide, and 6 feet in height ? 6. In 174964 cord feet, how many cords? 7. In 7645900 cubic inches, how many tons of hewn timber ? WINE OR LIQUID MEASURE. 111. Wine measure is used for measuring all liquids. TABLE. 4 gills, gi. make 1 pint, marked pt. 2 pints 1 quart, - qt. 4 quarts 1 gallon, - gal. 31 1 gallons - 1 barrel, - bar. or bbl. 42 gallons - 1 tierce tier, 63 gallons - 1 hogshead, hhd. 2 hogsheads 1 pipe pi. 2 pipes or 4 hogsheads 1 tun, tun. 111. What is measured by wine or liquid measure ? What are its denominations ? Repeat the table. What are the units of the scale ? What is the standard wine gallon? DENOMINATE NUMBERS. 107 gi. pt. qt. gal. bar. tier. hhd. pi. tun. 4 = 1 8 =2 =1 32 =8 =4 =1 1008 =252 =126 =311 -i 1344 =336 =168 =42 =1 2016 =504 =252 =63 =1$ =1 4032 =1008 =504 =126 =3 =2 = 1 8064 =2016 =1008 =252 =6 =4 = 2 =1 NOTE. The standard unit, or gallon of liquid measure, in the United States, contains 231 cubic inches. 1. How many gills in 4 pints ? How many pints in 3 quarts ? In 6 quarts ? In 9 ? In 10 ? 2. How many quarts in 2 gallons ? In 4 gallons ? In 6 gallons ? How many pints in 2 gallons ? In 5 ? 3. How many barrels in a hogshead ? How many in 4 hogsheads ? In 6 ? 4. How many quarts in 3 gallons? In 5 gallons? In 20? In a barrel how many ? In a hogshead how many ? EXAMPLES. 1. In 5 tuns 3 hogsheads 17 gallons of wine, how many gallons? OPERATION. btuns 3hhd. 17 gal. 4 23 63 76 139 2. In 1466 gallons, how- many tuns, &c. ? OPERATION. 63)1466 4)23 17 gal. 5 3 hhd. Ans. Stuns Bhhd. llgal. 14 66 gallons. 3. In 12 pipes 1 hogshead and 1 quart of wine, how many pints ? 4. In 10584 quarts of wine, how many tuns ? 5. In 201632 gills, how many tuns? 6 What will be the cost of 3 hogsheads, 1 barrel, 8 gal- lons, and 2 quarts of vinegar, at 4 cents a quart ? 108 REDUCTION OF ALE OR BEER MEASURE. 112. Ale or Beer Measure was formerly used for mea- suring ale, beer, and milk. TABLE, make 1 quart, marked qt. - 1 gallon, - - 1 barrel, - - 1 hogshead, 2 pints, pt. 4 quarts 36 gallons 54 gallons pt. 2 8 288 432 4 144 216 gal. bar. gal. bar. hhd. hhd. = 1 = 36 =1 = 54 =11 =1 NOTE. 1 gallon, ale measure, contains 282 cubic inches. 1. How many pints in 3 quarts ? How many in 5? 2. How many quarts in 3 gallons ? In 4 gallons ? In 9 ? EXAMPLES. 1 . How many quarts are there in hhd. 26ar. OPERATION. 4hhd. 26ar. Wgal. 8qt. li 4 4 86ar. 36 57 26 317 gal. 2. In 1271 quarts, how many hogsheads, &c. ? OPERATION. 4)1271 36)317 Zqt. Ans. hhd. 26ar. ZSgal. Zqt. 3. In^476ar. Ifigal. &qt., how many pints ? 4. In 27Md. 36ar. 25</a/. 3(?., how many pints ? 5. In 55832 pints, how many hogsheads ? 6. In 64972 quarts, how many barrels ? 112. For what is ale or beer measure used? iuatious ? Repeat tho table. What are its denotn- DENOMINATE NUMBERS. 109 DRY MEASURE. 113. Dry Measure is used in measuring all dry articles, such as grain, fruit, salt, coal, &c. TABLE. 2 pints, pt. 8 quarts - 4 pecks - 36 bushels - make 1 quart, marked 1 peck, - bushel, chaldron, bu. pk. bu. ch. ch. _ 16 =8 =1 64 =32 =4 2304 = 1152 = 144 1. How maty quarts in 2 pecks ? 2. How many pecks in 24 quarts ? 3. How many pecks in 6 bushels ? many bushels in 16 pecks ? In 32 ? = 1 = 36 = 1. In 5 ? In 8 ? In 32 ? In 64 ? In 8? In 12? How In 40? 4. How many bushels in 2 chaldrons ? In 3 ? In 4 ? NOTE. The standard bushel of the United States is the Win- chester bushel of England. It is a circular measure, 18 inches in diameter and 8 inches deep, and contains 2150s cubic inches, nearly. 2. A gallon, dry measure, contains 268 cubic inches. EXAMPLES. 1. How many quarts are there in 65c7i. 206w. 3pk. Iqt. ? OPERATION. 5c/i. 206w. Bpk. *lqt. 36 2. How many chaldrons, &c., in 75551 quarts? OPERATION. 8)75551 390 19T 4)9443 Iqt. 36)2360 Zplc. 2360 4 65 206w. 9443 8 Ans 65cA 20ta 3 Ic 7 t 75551 quarts. 113. What articles are measured by dry measure? What are its denominations? Repeat the table. What J3 the standard bushel? What arc the contents of a gallon? 110 REDUCTION OF 3. In 312 bushels, how many pints ? 4. In 5 chaldrons 31 bushels, how many pecks ? 5. In 17408 pints, how many bushels? 6. In 4220 pints, how many chaldrons ? AVOIRDUPOIS WEIGHT. 114. By this weight all coarse articles are weighed, such as hay, grain, chandlers' wares, and all metals except gold and silver. TABLE. 16 drams, dr. make 1 ounce, marked oz. 16 ounces 1 pound, lb. 25 pounds 1 quarter, - qr. 4 quarters - - 1 hundred weight, cwt. 20 hundred weight 1 ton, - T. qr. cwt. T. dr. oz. lb. 16 = 1 256 = 16 = 1 6400 = 400 = 25 25600 = 1600 = 100 = 4 =1 512000 = 32000 = 2000 = 80 = 20 = 1 NOTES. 1. The standard avoirdupois pound is the weight of 27.7015 cubic inches of distilled water. 2. By the old method of weighing, adopted from the English system, 112 pounds were reckoned for a hundred weight. But now, the laws of most of the States, as well as general usage, fix the hundred weight at 100 pounds. 3. The units of the scale, in passing from drams to ounces, are 16 ; from ounces to pounds, 16 ; from pounds to quarters, 25 ; from quarters to hundreds, 4 ; and from hundreds to tons, 20. 1. In 2oz., how many drams ? In 3 ? In 4 ? In 5. 2. In 4/6., how many ounces ? In 3 how many ? In 2 ? 3. In Qqr., how many hundred weight ? In bqr. ? 4. In Scwt., how many quarters ? How many in cwt. ? 5. In 60 hundred weight, how many tons ? In 80 ? 114. For what is avoirdupois weight used ? How is the table to be read ? How can you determine, from the second table, the value of any unit in units of the lower denominations ? DENOMINATE NUMBERS. Ill EXAMPLES. 1. How many pounds are there in 15T. Scurf. 3qr. ' 15/6. ? OPERATION. 15 T. Scwt. Sqr. 15/6. 20 308 cwtt 4 1235 qr. 25 ~6l80 5 /6. added. 2471 1 ten added. 30890 Ib. 2. In 30890 pounds, how many tons ? OPERATION. 25)30890 4)1235gr. 15/6. . Zqr. Scwt. Ans. 15 T. 8cut. 3qr. 15/6. 3. In 5T. Scurf. 3#r. 24/6. 13oz. 14dr., how many drams ? 4. In 28 T. 4curf. Iqr. 21/6., how many ounces? 5. In 2790366 drams, how many tons? 6. In 903136 ounces, how many tons? 7. In 3124446 drams, how many tons? 8. In 93 T. 13cwrf. 3qr. 8/6., how many ounces? 9. In 108910592 drams, how many tons ? 10. What will be the cost of 11 T. 17curf. Sqr. 24/6. of hay at half a cent a pound ? How much would that be a ton ? 11. What is the cost of 2T. 13cw;/. 3?r. 21/6. of beef at 8 cents a pound ? How much would that be a ton ? TROT WEIGHT. 115. Gold, silver, jewels, and liquors, are weighed by Troy weight. TABLE. 24 grains, gr. make 1 pennyweight, marked pwt. 20 pennyweights - 1 ounce - - - oz. 12 ounces - 1 pound - - Ib. gr. 24 480 5760 pwt. = 1 = 20 = 240 oz. i = 12 Ib. 1. 112 REDUCTION OF NOTES. 1. The standard Troy pound is the weight of 22.794377 cubic inches of distilled water. It is less than the pound avoirdupois. 2. The units of the scale, in passing from grains to penny- weights, are 24 ; from pennyweights to ounces, 20 ; and from ounces to pounds, 12. 1. How many grains in 2 pennyweights ? In 3 ? In 4 ? 2. How many pennyweights in 48 grains ? In 72 ? 3. How many ounces in 40 pennyweights ? In 60 ? 4. How many ounces in 4 pounds ? In 12 ? In 9? In 7 ? 5. How many pounds in 24 ounces ? In 36 ?" In 96 ? EXAMPLES. 1. How many grains are there in 16/6. lloz. Ibpwt. OPERATION. 16/6. lloz. 15pwt. 17or. 12 203 ounces. 20 4075 pennyweights 24 97817 grains. 2. In 97817 grains, how many pounds ? OPERATION. 24)97817 20)4075 pwt. 17or. 12) 203 oz. 15pwt. 16/6. lloz. Ans. 16/6. lloz. I5pwt. 11 gr. 3. In 25/6. 9oz. 20or., how many grains ? 4. In 6490 grains, how many pounds ? 5. In 148340 grains, how many pounds ? 6. In 117/6. 9oz. Ibpwt. ISgr., how many grains ? 7. In 8794pio/., how many pounds ? 8. In 6/6. 9oz. 21grr., how many grains ? 9. In 1/6. loz. Wpivt. 16#?\, how many grains t 10. A jewel weighing 2oz. \pwt. 18</r., is sold for half a dollar a grain : what is its value ? Notes. 1. What is the standard avoirdupois pound ? 2. What is a hundred weight by the English method? What is a hundred weight by the United States method ? '.>. Name the units of the scale in passing from one denomination to another. 115. What articles are weighed bv Troy weight ? What arc its de- nominations? Repeat the table? What is the standard Troy pound ? What arc the units of the scale, in passing from one unit to another ? DENOMINATE NUMBERS. 113 APOTHECARIES' WEIGHT. 110. This weight is used by apothecaries and physicians in mixing their medicines. But medicines are generally sold, in the quantity, by avoirdupois weight TABLE. 20 grains, gr. make 1 scruple, marked 3. 3 scruples - - 1 dram, - - - 3 8 drams - - - 1 ounce, - - - | . 12 ounces- - - 1 pound, - - - fi>. gr. 20 60 480 5760 3 1 3 24 288 .1 8 96 I __ = 12 = 1 NOTES. 1. The pound and ounces are the same as the pound and ounce in Troy weight. 2. The units of the scale, in passing from grains to scruples, are 20 ; in passing from scruples to drams, 3 ; from drams to ounces, 8 ; and from ounces to pounds, 12. 1. How many grains in 2 scruples ? In 3 ? In 4 ? In 6 ? 2. How many scruples in 4 drams ? In 7 drams ? In 5 ? 3. How many drams hi 5 ounces ? How many ounces in 32 drams ? EXAMPLES. 1. How many grains in > 8 63 23 OPERATION. 9fi> 8 3 63 12 23 116 ounces. 8 9d4 scruples. _3 2804 drams. 20 56092 grains. 2. In 56092 grains, how many pounds ? OPERATION. 20)56092 3)28043 ~8)9343 23 63 81 Am. 9fi> 8 | 63 23 REDUCTION OF 3. In 27 ft> 9 63 13, bow many scruples ? 4. In 94ft) 11 | 13, how many drams ? 5. 8011 scruples, how many pounds? 6. In 9113 drams, how many pounds ? 7. How many grains in 12ft> 9 73 23 8. In 73918 grains, how many pounds? MEASURE OF TIME. 117. TIME is a part of duration. The time in which the earth revolves on its axis is called a day. The time in which it goes round the sun is 365 days and 6 hours, and is called a year. Time is divided into parts according to the following TABLE. 60 seconds, sec. 60 minutes - 24 hours - 7 days 4 weeks - 13 wo. Ida. and 6/irs. ; or 365 da. Qhr. 12 calendar months - sec. 60 3600 86400 604800 m. = 1 = 60 1440 = 10080 nak e 1 minute, marked 1 hour, 1 day, 1 week, 1 month, m. hr. da. wk. mo. j- 1 Julian year, yr. - 1 year, yr. hr. da. wk. __ 1 24 = 1 168 =7 =1 yr. 31557600 = 525960 = 8766 = 365J =52 =1 NOTES. 1. The years are numbered from the beginning of the Christian Era. The year is divided into 12 calendar months, numbered from January : the dtays are numbered from the begin- ning of the month : hours from 12 at night and 12 at noon. . 31 - 31 . 30 - 31 . 30 . 31 Names. January,- - February, - March, - - April, - - - May, - . . June - - - No. - 1st. - 2d. - 3d. - 4th. - 5th. . 6th. No. i lays. 31 28 31 30 01 30 Names. July, - - - August, - - September, - October, - - November, - December, - No. . 7th. - 8th. - 9th. - 10th. - llth. - 12th. DENOMINATE NUMBERS. 115 2. The leogth of the tropical year is 365<J. 57ir. 48m. 4Ssec. nearly ; but in the examples we shall regard it as 365d. 6/w. 3. Since the length of the year is 365 days and 6 hours, the odd G hours, by accumulating for 4 years, make 1 day, so that every fourth year contains 366 days. This is called Bissextile or Leap Year. The leap years are exactly divisible by 4: 1872, 1876, 1880, are leap years. 4. The additional day, when it occurs, is added to the month of February, BO that this month has 29 days in the leap year. Thirty days hath September, April, June, and November ; All the rest have thirty-one, Excepting February, twenty-eight alone. 1. How many seconds in 4 minutes ? How many in 6 ? 2. How many hours in 3 days ? How many in 5 ? In 8 ? 3. How many days in 6 weeks ? In 8, how many ? 4. How many hours in 1 week ? How many weeks in 42da. ? EXAMPLES. 1. How many seconds in Qhr. ? OPERATION. 365da. 6/ir. 24 1466 730 2. How many days, &c. in 31557600 seconds? OPERATION. 60)31557600 60)525960 24)8766 365 6/ir. Ans. 365tfa. Qhr. 8766 60 525960x60 = 31557600 sec. 3. If the length of the year were 365da. 23/ir. 57m. 39sec., how many seconds would there be in 12 years? 4. In 126230400 seconds, how many years of 365 days? 5. In 756952018 seconds, how many years of 365 days ? 117. What are the denominations of time? How long is a year? How many days in a common year? How many days in a Leap year? How many calendar months in a year ? Name them, and the number of days in each. How many days has Februarv in the leap year ? How do you remember which of the months have 30 days, and which 31 ? 116 REDUCTION OF 6. In 285290205 seconds, how many years of 365da. 6Ar. each? 7. How many hours in any year from the 31st day of March to the 1st day of January following, neither day named being counted ? CIRCULAE MEASURE. 118. Circular measure is used in estimating latitude and longitude, and also in measuring the motions of the heavenly bodies. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees. Each degree is divided into 60 minutes, and each minute into 60 seconds. TABLE. 60 seconds' make 1 minute, marked '. 60 minutes 1 degree, - - . 30 degrees - 1 sign s. 12 signs or 360 - 1 circle c. 60 = 1 3600 =60 =1 108000 = 1800 =30 =1 1296000 = 21600 =360 =12 =1 1. How many seconds in 3 minutes ? In 4 ? In 5 ? 2. How many minutes in 6 degrees ? In 4 ? In 5 ? 3. How many degrees in 4 signs ? In 6 ? In 7 ? In 8 ? 4. How many degrees in 240 minutes ? In 720 ? How many signs in 90 ? In 150 ? In 180 ? EXAMPLES. 1. In 5s. 29 25', how many minutes ? 2. In 2 circles, how many seconds ? 3. In 27894 seconds, how many degrees, &c. ? 4. In 32295 minutes, how many circles, &c. ? 5. In 3 circles 16 20', how many seconds : 6. In 8s. 16 25", how many seconds ? 7. In 8589 seconds, how many degrees, &c. ? 118. For what is circular measure used? How is every circle sup- posed to be divided ? Repeat the table. DENOMINATE NUMBERS. 117 MISCELLANEOUS TABLE. 12 units, or things make 1 dozen. 12 dozen - 1 gross. 12 gross, or 144 dozen ' 1 great gross. 20 things - 1 score. 100 pounds - 1 quintal of fish. 196 pounds 1 barrel of flour. 200 pounds - 1 barrel of pork. 18 inches 1 cubit. 22 inches, nearly - 1 sacred cubit. 14 pounds of iron or lead - 1 stone. 21 J stones - 1 pig. 8 pigs 1 fother. BOOKS AND PAPER. The terms, folio, quarto, octavo, duodecimo, &c., indicate the number of leaves into which a sheet of paper is folded. A sheet folded in 2 leaves is called a folio. A sheet folded in 4 leaves " a quarto, or 4to. A sheet folded in 8 leaves " an octavo, or 8vo A sheet folded in 12 leaves " a 12mo. A sheet folded in 16 leaves " a 16mo. A sheet folded in 18 leaves " an 18mo A sheet folded in 24 leaves " a 24mo. A sheet folded in 32 leaves " a 32mo. 24 sheets of paper make 1 quire. 20 quires - - 1 ream. 2 reams - 1 bundle. 5 bundles 1 bale. MISCELLANEOUS EXAMPLES. 1. How many hours in 344wfc. Qda. llhr. ? 2. In 6 signs, how many minutes ? 3. In 15 tons of hewn timber, how many cubic inches ? 4. In 171360 pence, how many pounds? 5. In 1720320 drams, how many tons? 6. In 55799 grains of laudanum, how many pounds? 7. In 9739 grains, how many pounds Troy? 8. In 59/6. ISpwt. 5grr., how many grains ? 9. In .85 8s., how many guineas ? 10. In 346 E. F., how many Ells English ? i 118 KEDUCTION OF 11. In 3hhd. ISgal. 2qt., how many half-pints ? 12. In 12 T. Ibcwt. Iqr. 1Mb. 12dr., how many drams? 13. In 40144896 square inches, how many acres? 14. In 5760 grains Troy, how many pounds? 15. In 6 years (of 52 weeks each), 3>2wk. bda. 17/ir., how many hours ? 16. In 811480", how many signs ? 17. In 2654208 cubic inches, how many cords ? 18. In 18 tons of round timber, how many cubic inches ? 19. In 84 chaldrons of coal, how many pecks? 20. In 302 ells English, how many yards ? 21. In Qihhd. ISgal. 2qt. of molasses, how many gills ? 22. In 76 A IB. 8P., how many square inches? 23. In 15 19s. lid. 3/ar., how many farthings? 24. In 445577 feet, how many miles? 25. In 37444325 square inches, how many acres ? 26. If the entire surface of the earth is found to contain 791300159907840000 square inches, how many square miles are there ? 27. How many times will a wheel 16 feet and 6 inches in circumference, turn round in a distance of 84 miles ? 28. What will 28 rods, 129 square feet of land cost at $12 a square foot ? 29. What will be the cost of a pile of wood 36 feet long 6 feet high and 4 feet wide, at 50 cents a cord foot ? 30. A man has a journey to perform of 288 miles. He travels the distance in 12 days, travelling 6 hours each day : at what rate does he travel per hour ? 31. How many yards of carpeting 1 yard wide, will carpet a room 18 feet by 20? 32. If the number of inhabitants in the United States is 24 millions, how long will it take a person to count them, counting at the rate of 100 a minute ? 33. A merchant wishes to bottle a cask of wine containing 126 gallons, in bottles containing 1 pint each : how many bottles are necessary ? 34. There is a cube, or square piece of wood, 4 feet each way : how many small cubes of 1 inch each way, can be sawed from it, allowing no waste in sawing ? 35. A merchant wishes to ship 285 bushels of flax-seed in casks containing 7 bushels 2 pecks each : what number of casks are required ? DENOMINATE NUMBERS 119 36. How many times will the wheel of a car, 10 feet and 6 inches in circumference, turn round in going from Hartford to New Haven, a distance of 34 miles ? 37. How many seconds old is a man who has lived 32 years and 40 days ? 38. There are 15713280 inches in the distance from New York to Boston, how many miles ? 39. What will be the cost of 3 loads of hay, each weighing IScwt. 3qr. 24/6., at 7 mills a pound? ADDITION OF DENOMINATE NUMBERS. 119. Addition of denominate numbers is the operation of finding a single number equivalent in value to two or more given numbers. Such single number is called the sum. How many pounds, shillings, and pence in 4 8s. 9c?., 27 14s. lid., and 156 17s. lOd. ? ANALYSIS. We write the units of the same OPERATION. name in the same column. Add the column . s. d. of pence ; then 30 pence are equal to 2 shil- 489 lings and 6 pence : writing down the 6, carrying 9 * -. - , , the two to the shillings. Find the sum of the JJ 1 J iL shillings, which is 41 ; that is, 2 pounds and 1 shilling over. Write down 1*. ; then, carrying ^189 l s< g^ the 2 to the column of pounds, we find the sum to be 189 Is. 6d. NOTE. In simple numbers, the number of units of the scale, at any place, is always 10. Hence, we carry 1 for every 10. In denominate numbers, the scale varies. The number of units, in passing from pence to shillings, is 12 ; hence, we carry one for every 12. In passing from shillings to pounds, it is 20 ; hence, we carry one for every 20. In passing from one denomination to another, we carry 1 for so many units as are contained in the scale at that place. Hence, for the addition of denominate numbers, we have the following RULE. I. Set down the numbers so that units of the same name shall stand in the same column ; II. Add as in simple numbers, and carry from one de- nomination to another according to the scale. PROOF. The same as in simple numbers. 119. What is addition of denominate numbers? How do .you set down the numbers for addition ? How do you add ? How do you prove addition ? ^- ADDITION OF ( 8 } 173 13 87 17 75 18 d. 5 7* EXAMPLES. (2.) s d 705 17 3J 354 17 2j 175 17 3| (3.; s. 104 18 404 17 467 11 I d. 9| '4 25 17 4 87 19 71 597 14 *i 10 10 ii 52 12 7| 22 18 5 373 18 3 18 6 5 TROY WEIGHT. (4.) (5.) Ib. oz. pwt. gr. Ib. oz. pwt. gr. Ldd 100 10 19 20 171 6 13 14 432 6 5 391 11 9 12 80 3 2 1 230 6 6 13 7 9 94 7 3 18 11 10 23 42 10 15 20 8 9 31 21 APOTHECARIES' WEIGHT. (6.) (7.) (8.) ft) ! 3 3 gr. I 3 3 gr. 33 gr. 24 7 2 1 16 11 2 1 17 3 2 15 17 It 7 2 19 7 4 2 14 1 13 36 6 5 7 4 1 19 2 2 11 15 9 7 1 13 2 5 2 11 7 17 93419 10 1 2 16 5 2 14 AVOIRDUPOIS WEIGHT. (9.) (10.) cwt. qr. Ib. oz. dr. T. cwt. qr. Ib. oz. 14 2 14 9 15 12 1 10 10 13 2 20 1 15 71 8 2 6 93673 83 19 3 15 5 10 18 12 11 36 7 20 14 73232 47 11 2 2 11 6 1 19 8 1 63 5 2 19 7 4 , 3 15 5 12 13 1 14 9 12 2 13 9 7 5 10 DENOMINATE NUMBERS. 121 11. A merchant bought 4 barrels of potash of the following weights, viz. : 1st, 3cwt. 2qr. Mb. 12oz. 3dr. ; 2d, cwt. Iqr. 21/6. 4oz. ; 3d, cwt ; 4th, icwt. Qqr. 2/6. 15oz. 15dr. : what was the entire weight of the four barrels ? LONG MEASURE. L. 16 .<"< mi. fur. 2 7 i rd. yd. ft. 39 9 2 rd. 16 yd. ft. 9 2 171. 11 327 1 2 20 7 1 12 11 1 9 87 1 15 6 1 18 14 7 1 1 1 1 2 2 19 15 2 1 CLOTH MEASURE. (14.) E. Fl qr. 126 4 na. 4 (15.) yd. qr. 4 3 na. 2 E.E. 128 (16.) qr. na. 5 1 in. 3 65 3 1 5 4 1 20 3 1 2 72 1 3 6 1 19 1 4 1 157 2 3 25 2 2 15 3 1 2 LAND OR SQUARE MEASURE. (17.) (18.) Sq. yd. Sq.ft. Sq. in. M. A. R. P. Sq.yd 97 4 104 2 60 3 37 25 22 3 27 6 375 2 25 21 105 8 2 7 450 1 31 20 37 7 127 11 30 25 19 19. There are 4 fields, the 1st contains 12A 2P. 38P. ; the 2d, 4: A. IR. 26P. ; the 3d, 85 A QR. 19P. ; arid the 4th, 57 A IR. 2P. : how many acres in the four fields ? CUBIC MEASURE. (20.) (21.) (22.) Cu.yd. Cu.ft. Cu.in. C. S.ft. C. Cord ft. 65 25 1129 16 127 87 9 37 26 132 17 12 26 7 50 1 1064 18 119 16 6 22 19 17 37 104 19 5 122 ADDITION OF WINE OR LIQUID MEASURE. (23.) (24.) hhd. gal. qt. pt. tun. pi. hhd. gal. qt. 127 65 3 2 14 2 1 27 3 12 60 2 3 15 1 2 25 2 450 29 1 4 2 1 27 1 21 023 501 62 3 14 39 1 2 7 1 2 21 2 DRY MEASURE. (25.) (26.) ch. bu. pk. qt. pt. ch. bu. pk. qt. pt. 27 25 3 7 1 141 36 3 7 2 59 21 2 6 3 21 32 2 4 1 21271 85 9103 5 9182 10 4413 TIME. (27.) (28.) yr. mo. wk. da. hr. wk. da. hr. m. sec. * 4 11 3 6 20 8 8 14 55 57 3 10 2 5 21 10 7 23 57 49 5 8 1 4 19 20 6 14 42 01 101 9 3 7 23 6 5 23 19 59 55 8 4 6 17 2 2 20 45 48 CIRCULAR MEASURE OR MOTION. (29.) (30.) s. ' " s. ' " 5 17 36 29 6 29 27 49 7 25 41 21 8 18 29 16 8 15 16 09 7 09 04 58 NOTE. Since 12 signs make a circumference of a circle, we write down only the excess over exact 12's. APPLICATIONS IN ADDITION. 1. Add 46/6. 9oz. Ifywot. 16<?r., 87/6. lOoz. Gpwt. Ugr., 100/6. lOoz Wpwt. 10#r., and 56/6. Zpwt. 6gr. together. DENOMINATE NUMBERS. 123 2. What is the weight of forty-six pounds, eight ounces, thirteen pennyweights, fourteen grains ; ninety-seven pounds, three ounces ; and one hundred pounds, five ounces, ten pen- nyweights and thirteen grains ? 3, Add the following together: 29 T. Ibcwt. Iqr. 14/6. 12oz. Mr., IScwt. 3?r, lib., 50 1 7 . 3?r 4oz., and 2T. Iqr. 4 What is the weight of 39 T. Wcwt. 2?r. 2/6. 15oz. I2dr., llcwt. 6/6., I2cwt. 3?r., and 2?r. Sib. Mr.l 5. What is the sum of the following : 314^4. 2E. 39P. 200s7. ft- 136s?. in., UA. IE. 20P. 10s?. ft., BE. 36P. and 4 A. IE. 16P.? 6. What is the solid content of 64fons 33/2. 800m., Qtons 1200m., 25/35., 700m., and 95tes 31/fc 1500m. 7. Add together, 966u. 3p&. 2qt. Ipt., 466w. 3pfc. 1?. Ipt., 2pk. Iqt. Ipt. and 236w. 3p&. 4?. lp. 8. What is the area of the four following pieces of land ; the first containing 20 A. BE. 15P. 250s?. ft. 116s?. in. ; the second, 19A IE. 39P. ; the third, 2P. 10P. 60s?, ft. ; and the fourth, 5 A. 6P. 50s?. in. ? 9. A farmer raised from one field 37Zw. Ipk. 3qt. of wheat ; from a second, 416w. 2pk. 5?. of barley ; from a third, 356w. Ipk. 3qt. of rye ; from a fourth, 436w. 3pk. Iqt. of oats ; how much grain did he raise in all ? 10. A grocer received an invoice of 4hhd. of sugar ; the first weighed llcwt. 15/6. ; the second, 12cwt. 3?r. 15/6. ; the third, Scwt. Iqr. 16/6. ; the fourth, I2cwt. Iqr. : how much did the four weigh ? 11. A lady purchased 32?/ds. 3?rs. of sheeting ; 31yds. Iqr. of shirting ; llyds. 2?rs. of linen ; and Qyds. 2??*s. of cambric : what was the whole number of yards purchased ? 12. Purchased a silver teapot weighing 23oz. llpivt. llgr. ; a sugar bowl, weighing 8oz. ISpwt. l$gr. a cream pitcher, weighing 5oz. ll^r. : what was the weight of the whole ? 13. A stage goes one day, 87m. Qfur. 24rd. ? the next, 75??i. 3/wr. 17r^. ; the third, 80m. Ifur. Wrd. ; the fourth, 78m. 5/*wr. : how far does it go in the four days ? 14. Bought three pieces of land ; the first contained 17 acres IE. 35?'rf. ; the second, 36 acres 2E. 2lrd. ; and the third, 46 acres QE. 37rd. : how much land did I purchase ? 124: SUBTRACTION OF SUBTRACTION OF DENOMINATE NUMBERS. 120. The difference between two denominate numbers is such a number as added to the less will give the greater. SUBTRACTION is the operation of finding this difference. I. What is the difference between 27 16s Sd and 19 17s. 9df.? ANALYSIS. We cannot take 9rf. from Sd. ; OPERATION. we therefore add to the upper number as many 20 12 units as are contained in the scale, and at the x** IDS. 8a. same time add 1, mentally, to the next higher 19 17 9 denomination of the subtrahend. We then say, To rr~ 9 from 20 leaves 11. Then, as we cannot sub- tract 18 from 1C,' we add 20 and say, 18 from 36 leaves 18. Now, as we have taken 1 pound=20 shillings, from the pounds, and added it to the shillings, there are but 26 pounds left. We may then say, 19 from 26 leaves 7, or 20 from 27 leaves 7. The lat- ter is the easiest in practice. The first step is called borrowing, the second, carrying : hence, RULE. I. Set down the less number under the greater, placing units of the same value in the same column. II. Begin with the lowest denomination, and subtract as in simple numbers, borrowing and carrying for each operation according to the scale. PROOF. The same as in simple numbers. EXAMPLES. (1.) (2-) A. E. P. T. cwt. qr. Ib. From - 18 3 28 4 12 3 20 Take - 15 2 30 ) 2 18 _ 3 1) Remainder ~3 (T~38 ) 1 14 19 ) Proof - liTir~28 4 12 3 20 (3.) (4 ) Ib. oz. pwt. gr. Ib. oz. pwt. gr. From - 273 18 9 10 Take - 98 10 18 21 9 10 15 20 Remainder DENOMINATE NUMBERS. 125 (5.) T, cwt. qr. Ib. oz. From - 7 14 1 3 6 Take - 2 6 3 4 11 Remainder T. hhd. gal. qt. pt. From - 151 3 50 3 2 Take - 27 2 54 3 2 Remainder (6.) cwt. qr. Ib. oz. dr. 14 2 12 10 8 6 3 16 15 3 (8.) yr. wk. da. hr. ' 95 25 4 20 45 50 80 30 6 23 46 56 TIME BETWEEN DATES. 121. To find the time between any two dates. 1. What time elapsed between July 5th, 1848, and August 8th., 1850 ? OPERATION. yr. mo. da. 1850 8 8 1848 7 5 213 NOTE. In the first date, the number of the year is 1848 ; the number of the month 7, and the number of the day, 5. In the second date, the number of the year is 1850, the number of the month 8, and the number of the day, 8. Hence, to find the time between two dates : Write the numbers of the earlier date under those of the later, and subtract according to the preceding rule. NOTE. 1. In finding the difference between dates, as in casting interest, the month is regarded as the twelfth part of a year, and as containing 30 days. 2. The civil day begins and ends at 12 o'clock at night. 2. What is the difference of time between March 2d, 1847, and July 4th, 1856? 3. What is the difference of time between April 28th, 1834, and February 3d, 1856 ? 4. What time elapsed between November 29th, 1836, and January 2d, 1854 ? 120. What is the difference between two denominate numbers? Give the rule for subtraction. How do you prove subtraction ? 131. Give the rule for finding the difference between two date*- How is the month reckoned ? At what time docs a civil day begin ? 126 SUBTRACTION OF 5. What time elapsed between November 8th, at 1 1 o'clock A.M., 1847, and December 16th, at 4 o'clock, P.M., 1850 ? OPERATION. ANALYSIS. The hours are numbered 1/r vnn fj n i> r _ y / //tC/. tit/. /If . from 12 at night, when the civil day begins. 1359 10 IA i/ The numbers of the years, months, days 184 * and hours are used. 3185 6. What time elapsed between October 9th, at 11 P.M., 1840, and February 6th, at 9 P.M., 1853 ? 7. Mr. Johnson was born September 6th, 1771, at 9 o'clock A.M., and his first child November 5th, 1801, at 9 o'clock P.M. : what was the difference of their ages ? APPLICATIONS IN ADDITION AND SUBTRACTION. 1. From 38mo. 2wk. Zda. 7/ir. 10m., take lOmo. Zwk. 2da. Whr. 50m. 2. From 176t/r. 8mo. 3wh 4da., take 91yr. 9mo. 3. From 3, take 3s. 4. From 2/6. take 20#r. Troy. 5. From 8R, take lft> 1 3 23 23. 6. From 9T. r take IT. lewt. 2qr. 20/6. 15o2. 7. From 3 miles, take 3/wr. 19rd. 8. The revolution commenced April 19th, 1775, and a general peace took place January 20, 1783 : how long did the war continue ? 9. America was discovered by Columbus, October 11, 1492 : what was the length of time to July 25, 1855 ? 10. I purchased 167/6. 8oz. IGpwt. lOgrr. of silver, and sold 98/6. lOoz. I2frwt. Wgr. : how much had I left? 11. I bought 19T. llcwt. Zqr. 2/6. 12oz., 12c?r. of old ,'ron, and sold 17 T. IScwt. 2^r. 19/6. 14oz. lOc^r. : what had I left ? 12. I purchased lOlIbll? ^3 23 19pr. of medicine, and sold 17ft>2333 1& bgr.: how much remained un- sold? 13. From 46?/d. Iqr. 3na., take 42^. 3qr. Ina. 2m. 14. Bought 7 cords of wood, and 2 cords 78 feet having been stolen, how much remained ? DENOMINATE NUMBERS. 157 .5. A owes B 100 : what will remain due after he has paid him 25 3s. 6J<*. ? 16. A farmer raised 136 bushels of wheat ; if he sells 496w. 2p. Iqt. Ipt., how much will he have left? 17. From 174/iM. Wgal. Iqt. Ipt. of beer, take SQhhd. 17 gals. 2qt. Ipt. * 18. A farmer had 5766w. Ipk. %qt. of wheat ; he sold 1396w. 2p&. 3qt. Ipt. : how much remaiued unsold? 19. A merchant bought Vlcwt. 2qr. 14/6. of sugar, of which he sold at one time 3cwt. Zqr. 20/6. ; at another Qcwt. Iqr. 5/6. : how much remained unsold ? 20. Sold a merchant one quarter of beef for 2 7s. 9d ; one cheese for 9s Id. ; 20 bushels of corn for 4 10s. lid. ; and 40 bushels of wheat for 19 12s. 8Jd. : how much did the whole come to ? 21. Bought of a silversmith a teapot, weighing 3/6. 4oz. Qpivt. 2lgr. ; one dozen of silver spoons, weighing 2/6. loz. Ipwt. ; 2 dishes weighing 16/6. lOoz. ISpwt. IQgr. : how much did the whole weigh ? 22. Bought one hogshead of sugar weighing $cwt. 3qr. 2/6. 14oz. ; one barrel weighing 3cwt. Iqr. 2/6., and a second barrel weighing Scwt. Qqr. lib. 4oz. : how much did the whole weigh? 23. A merchant buys two hogsheads of sugar, one weigh- ing Scwt. 3qr. 21/6., the other 9cwt. 2qr. 6/6. ; he sells two barrels, one weighing 3cwt. Iqr. 12/6. 14oz., the other, Zcwt. Bqr. 15/6. 6oz. : how much remains on hand ? 24. A man sets out upon a journey and has 200 miles to travel ; the first day he traveled 9 leagues 2 miles 7 furlongs 30 rods ; the second day 12 leagues 1 mile 1 furlong ; the third day 14 leagues ; the fourth day 15 leagues 2 miles ^ 5 furlongs 35 rods : how far had he then to travel ? 25. A farmer has two meadows, one containing A. ZR. 37P., the other contains 10A 2R. 25P. ; also three pas- , tures, the first containing 12^4. IE. IP. ; the second con-' taining 13A BE., and the third &A. IE. 39P. : by how many acres does the pasture exceed the meadow land ? 26. Supposing the Declaration of Independence to have been published at precisely 12 o'clock on the 4th of July, 1776, how much time elapsed to the 1st of January, 1833, at 25 minutes past 3, T.M. ? 128 MULTIPLICATION OF MULTIPLICATION OF DENOMINATE NUMBERS. 122. MULTIPLICATION of denominate numbers is the opera- tion of multiplying a denominate number by an abstract number. I. A tailor has 5 pieces of cloth each containing 6yd~ %qr. 3na. : how many yards are there in all ? ANALYSIS. In all the pieces there are 5 OPERATION. times as much as there is in 1 piece. If in yd. or. na. 1 piece each denomination be taken 5 times, it o 3 the result will be 5 times as great as the multi- plicand. Taking each denomination 5 times, we have 30#d. lO^r. 15?ia. 30 10 15" But, instead of writing the separate products, 33 1 3 we begin with the lowest denomination and say, 5 times 3na. are 15na. ; divide by 4, the units of the scale, write down the remainder 3fta., and reserve the quotient Sgr. for the next product. Then say, 5 times 2qr. are 10r., to which add the %qr. making 13gr. Then divide by 4, write down the remainder 1, and reserve the quotient 3 for the next product. Then say, 5 times 6 are 30, and 3 to carry are 33 yards : hence, RULE. I. Write down the denominate number and set the multiplier under the lowest denomination. II. Multiply as in simple numbers, and in passing from one denomination to another, divide by the units of the scale, set down the remainder and carry the quotient to the next product. PROOF. The same as in simple numbers. 17 CM. s. d 15 9 .far. 6 EXAMPLES. T. c?r/. 10 (2. * :>* 2 oz. 12 7 106 14 10 (3.) m.fur. rd. 9 3 20 2 3 *? 6 10 8. 9 9 19 (4.) 27 4 35 3 132. What is multiplication of denominate numbers? Give the rule. How do you prove multiplication ? DENOMINATE NUMBERS. 129 (5.) (6.) yr. mo. da. hr. T. cwt. qr. Ib. oz. dr. 6 5 15 18 6 12 3 20 12 9 5 8 7. A farmer has 11 bags of corn, each containing 26w. Ipk. 3qt. : how much corn in all the bags ? 8. How much sugar in 12 barrels, each containing 3cw 3qr. 2/6. ? 9. In 7 loads of wood, each containing 1 cord and 2 cord feet, how many cords ? 10. A bond was given 21st of May, 1825, and was taken up the 12th of March, 1831 ; what will be the product, if the time which elapsed from the date of the bond till the day it was taken up, be multiplied by 3 ? 11. What is the weight of'l dozen silver spoons, each weighing 3oz. Spwt. ? 12. What is the weight of 7 tierces of rice, each weighing 5cwt. 2qr. 16/6.? ' 13. Bought 4 packages of medicine, each containing 3fi> 4^ 63 13 16#r. : what is the weight of all ? 14. How far will a man travel in 5 days at the rate of 24mi. 4/ur. krd. per day ? 15. How much land is there in 9 fields, each field contain- ing 12^. IK 25P.? 16. How many yards in 9 pieces, each 29 yd. 2qr. 3na. ? 17. If a vessel sails 5L. 2>mi. 6fur. SQrd. in one day, how far will it sail in 8 days ? 18. How much water will be contained in 96 hogsheads, each containing QZgal. Iqt. Ipt. Igi. ? NOTE. When the multiplier is a composite number, and the factors do not exceed 12, multiply by the factors in succession. In the last example 96=12 x 8. 19. If one spoon weighs 3oz. 5pwt. 15<?r. what is the weight of 120 spoons? i 20. If a man travel 249m. 7/itr. 4rd. in one day, how far will he go in one month of 30 days? 21. If the earth revolve 15' of space per minute of tune, how far does it revolve per hour ? 22. Bought 90/i/id. of sugar, each weighing IZcwt. Zqr. . : what was the weight of the whole? 130 DIVISION OF 23. What is the cost of 18 sheep, at 5s. 9|d. apiece ? 24. How much molasses is contained in 2bhhd. each hogs- head having ftlgal. \qt. Ipt. ? 25. How many yards of cloth in 36 pieces, each piece con- taining %5yd. 3qr. ? 26. A farmer has 18 lots, and each lot contains 41 A 2#. IIP. : how many acres does he own? 21. There are three men whose mutual ages are 14 times 2Qyr. 5mo. 3wk. Qda. : what is the sum of their ages? 28. Bought 90/i/id. of sugar, each weighing 12cwt. 2qr. 14lb. ; what is the weight of the whole ? 29. If a vessel sail 49ml Qfur. 8rd. in one day, how far will she sail in one month of 30 days ? 30. Suppose each of 50 farmers to raise 125m. 3pk. 6qt. of grain : how much do they all raise ? 31. If a steam ship, in crossing the Atlantic, goes 211mi. 4/wr. 32rd. a day, how far will she go in 15 days? 32. If 1 horse consume 2 tons Iqr. 20/6. of hay in a winter, how much will 36 horses consume? 33. How much cloth will clothe a company of 48 men, if it takes 5yd. 3qr. 2na. to clothe one man ? NOTE. Each denomination may Be multiplied by the multiplier, separately, and the results reduced and added. DIVISION OF DENOMINATE NUMBERS. 123. DIVISION of denominate numbers is the operation of dividing a denominate number into as many equal parts as there are units in the divisor. 1. Divide 25 15s. 4d. by 8. ANALYSIS. We first say 8 into 25, 3 times OPERATION. and 1 or 20s. over. Then, after adding the 8)d25 15s. la 15s. we say, 8 into 35, 4 times and 3s. over. ^ -j F~? Then, reducing the 3*. to pence and adding in the 4rf., we say, 8 into 40, 5 times. 123. What is division of denominate numbers? Give the rule for division. How do you prove division ? How do you divide when the divisor is a composite number ? What will be the unit of each quo- tient figure ? DENOMINATE NUMBERS. 131 OPERATION. t366w. 2. Divide 366tt. 3pfc. Iqt. by 7. ANALYSIS. In this example we find that 7 is contained in 36 bushels 5 times and 1 bushel over. Reducing this to pecks, and adding 3 pecks, gives 7 pecks, which contains 7, 1 time and no remainder. Multiplying by 8 quarts and adding, gives 7 quarts to be divided by 7. 7)7(lqt Ans. bbu. \pk. Iqt. Hence, for the division of denominate numbers we have the following RULE. I. Begin with the highest, denomination and divide as in simple numbers : II. Reduce the remainder, if any, to the next lower de- nomination, and add in the units of that denomination for a new dividend. III. Proceed in the same manner through all the denomi- nations. PROOF. By multiplication, as in simple numbers. NOTES. 1. If the divisor is a composite number, we may divide by the factors in succession, as in simple numbers. 2. Each quotient figure has the same unit as the dividend from which it was derived. 3. If the divisor is greater than 12 and not a composite number, the operation is the same as long division. EXAMPLES. (1.) (2.) T. cwt. qr. Ib. A. It. P. 7)1 19 2 12 9)113 3 25 Quotient. " 5216 122 25 (3.) (4.) L. mi. fur. rd. bu. pk. qL 8)47 1 7 8 11)25 3 1 Quotient. 132 DIVISION OF Divide the following : 5. l*lcwt. Qqr. 2/6. 6oz. by 7. 6. 49*/d. 3?r. 3/m. by 9. 7. 131A 1,8. by 12. 8. 1138 12s. 4a. by 53. 9. TOT. 17cwtf. 7/6. by 79. 10. 276u. Spyfc. 7^. by 84. 11. Bought 65 yards of cloth for which I paid 72 14s. . : what did it cost per yard? 12. If 15 loads of hay contain 35 T. 5cwt., what is the weight of each load ? 13. If a man, lifting 8 times as much as a boy, can raise 201/6. 12oz., how much can the boy lift? 14. If a vessel sail 25 42' 40" in 10 days, how far will she sail in one day ? 15. Divide Vhhd. ZZgal. 2qt. by 12. 16. What is the quotient of 656w. Ipk. 3qt. divided by 12? 17. In 4 equal packages of medicine there are 13B> 7 3 23 13 4gr. ; how much is there in each package ? 18. In 25hhd. of molasses, the leakage has reduced the whole amount to 1534gra/. \qt. \pt. : if the same quantity has leaked out of each hogshead, how much will each hogs- head still contain ? 19. In 9 fields there are 113A 37?. 25P. of land : if the fields contain an equal amount, how much is there in each field? 20. If in 30 days a man travels 746mi. 5/wr., travelling the same distance each day, what is the length of each day's journey ? 21. Suppose a man had 98/6. 2oz. Wpwt. 6gr. of silver ; how much must he give to each of 7 men if he divides it equally among them? 22. When J75#a/. 2qt. of beer are drank in 52 weeks, how much is consumed in one week ? 23 A rich man divided 1686w. Ipk. Qqt. of corn among 35 poor men : how much did each receive ? 24. In sixty-three barrels of. sugar there are 7T. 16cwtf. 3qr. 12/6. : how much is there in each barrel ? 25. A farmer has a granary containing 232 bushels 3 ks 7 quarts of wheat, and he wishes to put it in 105 bags : ow much must each bag contain ? 26. If 90 hogsheads of sugar weigh 56 T. Hcwt. Zqr. 15/6, what u the weight of 1 hogshead ? DENOMINATE NUMBERS. 133 27. One hundred and seventy-six men consumed in a week IScwt 2qr. 15/6. 6oz. of bread : how much did each man consume ? 28. If the earth revolves on its axis 15 in 1 hour, how far does it revolve in 1 minute ? 29. If 59 casks contain 44Md. ttgal. 2qt. Ipt. of wine, what are the contents of one cask ? 30. Suppose a man has 246ml Qfur. 36rd. to travel in 12 days : how far must he travel each day? 31. If I pay 12 14s. 5d 3/ar. for 35 bushels of wheat, what is the price per bushel ? 32. A printer uses one sheet of paper for every 16 pages of an octavo book : how much paper will be necessary to print 500 copies of a book containing 336 pages, allowing 2 quires of waste paper in each ream ?* 33. A man lends his neighbor 135 6s. 8d., and takes in part payment 4 cows at 5 8s. apiece, also a horse worth 50 : how much remained due ? 34. Out of a pipe of wine, a merchant draws 12 bottles, each containing 1 pint 3 gills ; he then fills six 5-gallon demi- johns ; then he draws off 3 dozen bottles, each containing 1 quart 2 gills : how much remained in the cask ? 35. A farmer has 6 T. Scivt. 2qr. 14/6. of hay to be re- moved in 6 equal loads : how much must be carried at each load? 36. A person at his death left landed estate to the amount of 2000, and personal property to the amount of 2803 17s. 4c?. He directed that his widow should receive one-eighth of the whole, and that the residue should be equally divided among his four children : what was the widow and each child's portion ? 37. If a steamboat go 224 miles in a day, how long will it take to go to China, the distance being about 12000 miles? 38. How long would it take a balloon to go from the earth to the moon, allowing the distance to be about 240000 miles, the balloon ascending 34 miles per hour ? * In packing and selling paper, the two outside quires of every ream are regarded as waste, and each of the remaining quires contains 34 perfect sheets: hence, in this example, the waste "paper is considered as belonging only to the entire reams. 134 LONGITUDE AND TIME. LONGITUDE AND TIME. 124. The circumference of the earth, like that of other circles, is divided into 360, which are called degrees of lon- gitude. 125. The sun apparently goes round the earth once in 24 hours. This time is called a day. Hence, in 24 hours, the sun apparently passes over 360 of longitude ; and in 1 hour over 360 -=-24 = 15. 126. Since the sun, in passing over 15 of longitude, re- quires 1 hour or GO' of time, 1 will require 60'-=- 15 = 4= minutes of time ; and V of longitude will be equal to one sixteenth of 4' which is 4" : hence, 15 of longitude require 1 hour 1 of longitude requires 4 minutes. 1' of longitude requires 4 seconds. Hence, we see that, 1. If the degrees of longitude be multiplied by 4, the pro- duct will be the corresponding time in minutes. 2. If the minutes in longitude be multiplied by 4, the pro- duct will be the corresponding time in seconds. 127. When the sun is on the meridian of any place, it is 12 o'clock, or noon, at that place. Now, as the sun apparently goes from east to west, at the instant of noon, it will be past noon for all places at the east, and before noon for all places at the west. If then, we find the difference of time between two places, and know the exact time at one of them, the corresponding time at the other will be found by adding their difference, if that the other be east, or by subtracting it if west. 124. How is the circumference of the earth supposed to be divided ? 125. How does the sun appear to move ? What is a day ? How far does the sun appear to move in 1 hour ? 126. How do you reduce degrees of longitude to time ? How do you reduce minutes of longitude to time ? 127. What is the hour when the sun is on the meridian ? When the sun is on the meridian of any place, how will the time be for all places cast? How for all places west? If you have the difference of time, how do you find the time V LONGITUDE AND TIME. 135 1. The longitude of New York is 74 1' west, and that of Philadelphia 75 10' west : what is the difference of longi- tude and what their difference of time ? 2. At 12 M. at Philadelphia, what is the time at New York? 3. At 12 M. at New York, what is the time at Philadelphia ? 4. The longitude of Cincinnati, Ohio, is 84 24' west : what is the difference of time between New York and Cin- cinnati ? 5. What is the time at Cincinnati, when it is 12 o'clock at New York? 6. The longitude of New Orleans is 89 2' west : what time is at New Orleans, when it is 12 M. at New York ? 7. The meridian from which the longitudes are reckoned passes through the Greenwich Observatory, London : hence, the longitude of that place is : what is the difference of time between Greenwich and New York ? 8. What is the time at Greenwich, when it is 12 M. at New York? 9. The longitude of St. Louis is 90 15' west : what is the time at St. Louis, when it is 3/i. 25m. P.M. at New York ? 10. The longitude of Boston is 71 4' west, and that of New Orleans 89 2' west : what is the time at New Orleans when it is 7 o'clock 12??i A.M. at Boston ? 11. The longitude of Chicago, Illinois, is 87 30' west : what is the time at Chicago, when it is 12 M. at New York? PROPERTIES OF NUMBERS. COMPOSITE AND PRIME NUMBERS. 128. An Integer, or whole number, is a unit or a collection of units. 129. One number is said to be divisible by another, when the quotient arising from the division is a whole number. The division is then said to be exact. NOTE. Since every* number is divisible by itself and 1, the term divisible will be applied to such numbers only, as have other divisors. 128. What is an Integer ? 136 PROPERTIES OF NUMBERS. 130. Every divisible number is called a composite number, (Art. 54), and any divisor is called & factor: thus, 6 is a com- posite number, and the factors are 2 and 3. 131. Every number which is not divisible is called a prime number : thus, 1, 2, 3, 5, 7, 11, &c. are prime numbers. 132. Every prime number is divisible by itself and 1 ; but since these divisors are common to all numbers, they are not called factors. 133. Every factor of a number is either prime or compo- site : and since any composite factor may be again divided, it follows that, Any number is equal to the product of all its prime factors. For example, 12=: 6 x 2 ; but 6 is a composite number, of which the factors are 2 and 3 ; hence, 12=2 x 3 x 2 ; also, 20=10 x 2=5 x 2 x 2. Hence, to find the prime factors of any number, Divide the number by any prime number that will exactly divide it : then divide the quotient by any prime number that will exactly divide it, and so on, till a quotient is found which is a prime number ; the several divisors and the last quotient will be the prime factors of the given number. ' NOTE. It is most convenient, in practice, to use the least prime number, which is a divisor. 1. What are the prime factors of 42 ? OPERATION. ANALYSIS. Two being the least divisor 2)42 that is a prime number, we divide by it, giv- o\ 91 ing the quotient 21, which we again divide o)4L by 3, giving 7: hence, 2, 3 and 7 are the 7 prime factors. 2x3x7 = 42. 129. When is one number divisible by another ? By what is every number divisible ? Is 1 called a divisor ? 130. What is a composite number ? What is a factor ? 131. What is a prime number ? 132. By what divisors is every prime number divided ? 133. To what product is every number equal? Give the rule for finding the prime factors of a number. What number is it most conve- nient to use as a divisor ? PRIME FACTORS. 137 What arc the prime factors of the following numbers ? 1. Of the number 9 ? 2. Of the number 15? 3. Of the number 24 ? 4. Of the number 16? 5. Of the number 18 ? 6. Of the number 32 ? 7. Of the number 48 ? 8. Of the number 56? 9. Of the number 63 ? 10. Of the number 76? NOTE. The prime factors, when the number is small, may generally be seen by inspection. The teacher can easily multiply the examples. 134. When there are several numbers whose prime factors are to be found, Find the prime factors of each and then select those factors which are common to all the numbers. 11. What are the prime factors common to 6, 9 and 24 ? 12. What are the prime factors common to 21, 63 and 84? 13. What are the prime factors common to 21, 63 and 105 ? 14. What are the common factors of 28, 42 and 70 ? 15. What are the prime factors of 84, 126 and 210 1 16. What are the prime factors of 210, 315 and 525 ? 135. DIVISIBILITY OF NUMBERS. 1. 2 is the only even number which is prime. 2. 2 divides every even number and no odd number. 3. 3 divides any number when the sum of its figures is di- visible by 3. 4. 4 divides any number when the number expressed by the two right hand figures is divisible by 4. 5. 5 divides every number which ends in or 5. 6. 6 divides any even number which is divisible by 3. 7. 10 divides any number ending in 0. GREATEST COMMON DIVISOR. 130. The greatest common divisor of two or more num- bers, is the greatest number which will divide each of them, separately, without a remainder. Thus, 6 is the greatest common divisor of 12 and 18. 134. How do you find the prime factors of two or more numbers ? 138 COMMON DIVISOR. NOTE. Since 1 divides every number, it is not reckoned among the common divisors. 137. If two numbers have no common divisor, they are called prime with respect to each other. 138. Since a factor of a number always divides it, it fol- lows that the greatest common divisor of two or more num- bers, is simply the greatest factor common to these numbers. Hence, to find the greatest common divisor of two or more numbers, I. Resolve each number into its prime factors. II. The product of the factors common to each result will be the greatest common divisor. EXAMPLES. 1. What is the greatest common divisor of 24 and 30 ? ANALYSIS. There are four prime OPERATION. factors in 24, and 3 in 30 : the factors 24 = 2x2x2x3 2 and 3 are common : hence, 6 is the 30 = 2 X 3 X 5 greatest common divisor. 2 X ^(> com. divisor. 2. What is the greatest common divisor of 9 and 18 ? , 3. What is the greatest common divisor of 6, 12, and 30 ? 4. What is the greatest common divisor of 15, 25 and 30 ? 5. What is the greatest common divisor of 12, 18 and 72 ? 6. What is the greatest common divisor of 25, 35 and 70 ? 7. What is the greatest common divisor of 28, 42 and 70 ? 8. What is the greatest common divisor of 84, 126 and 210? 139. When the numbers are large, another method of find- ing their greatest common divisor is used, which depends ou the following principles : 135. What even number is prime ? What numbers will 2 divide ? What numbers will 3 divide ? What numbers will 4 divide ? 5 ? 6 ? 10? 136. What is the greatest common divisor of two or more numbers ? 137. When are two numbers said to be prime with respect to each other? 138. What is the greatest factor of two numbers ? How do you find the greatest common divisor of two or more numbers ? PROPERTIES OF NUMBERS. 139 1. Any number which willdividetwo numbers separately, will divide their sum ; else, we should have a whole number equal to a proper fraction. 24+27=51 2. Any number which will divide two numbers separately, ivill divide their difference; and any number which will divide their differ- 51 27 = 24 ence and one of the numbers, will divide the other ; else, we should have a whole number equal to a proper fraction. 1. */ What is the greatest common divisor of 27 and 51 ? Divide 51 by 27 ; the quotient is 1 and the remainder 24 ; then divide the preceding divisor 27 by the re- OPERATION. mainder 24 : the quotient is 1 and the re 27)51(1 mainder 3 : then divide the preceding 27 divisor 24 by the remainder 3 ; the quo- tient is 8 and the remainder 0. 24 ) 27 ( 1 Now, since 3 divides the difference 3, and also 24, it will divide 27, by principle 3)24(8 2d ; and since 3 divides the remainder 24, 04 and 27, it will also divide 51 : hence it is a common divisor of 27 and 51 ; and since it is the greatest com- mon factor, it is their greatest common divisor. Since the above reasoning is as applicable to any other two numbers as to 27 and 51, we have the following rule : Divide the greater number by the less, and then divide the preceding divisor by the remainder, and so on, till nothing re- mains : the last divisor will be the greatest common divisor. EXAMPLES. 1. What is the greatest common divisor of 216 and 408 ? 2. Find the greatest common divisor of 408 and 740. 3. Find the greatest common divisor of 315 and 810. 4. Find the greatest common divisor of 4410 and 5670. 5. Find the greatest common divisor of 3471 and 1869. 6. Find the greatest common divisor of 1584 and 2772. NOTE. If it be required to find the greatest common divisor of more than two numbers, first find the greatest common divisor of 139. When the numbers are large, on what principles docs the oper- ation of finding the greatest common divisor depend ? What is the rule for finding it ? 140 COMMON MULTIPLE* two of them, then of that common divisor and one of the remain ing numbers, and so on for all the numbers ; the last common divisor will be the greatest common divisor of all the numbers. 7. What is the greatest common divisor of 492, 744 and 1044? 8. What is the greatest common divisor of 944, 1488, and 2088? 9. What is the greatest common divisor of 216, 408 and 740? 10. What is the greatest common divisor of 945 1560 and 22683 ? LEAST COMMON MULTIPLE. 140. The common multiple, of two or more numbers, is any number which will exactly divide. The least common multiple of two or more numbers, is the least number which they will separately divide without a re- mainder. NOTES. 1. If a dividend is exactly divisible by a divisor, it can be resolved into two factors, one of which is the divisor and the other the quotient. 2. If the divisor be resolved into its prime factors, the cor- responding factor of the dividend may be resolved into the same factors : hence, the dividend will contain every prime factor of the divisor. 3. The question of finding the least common multiple of several numbers, is therefore reduced to finding a number which shall con- tain all their prime factors and none others. 1. Let it be required to find the least common multiple of 6, 8 and 12. ANALYSIS. We see, from inspec- OPERATION. tion, that the prime factors of 6 are 2x3 2x2x2 2x2x3 2 and 3 : of 8 ; 2, 2 and 2 : and 6 8 12 of 12 ; 2, 2 and 3. Every number that is a prime factor must appear in the least com- mon multiple, and none others: hence, it will contain all the prime 140. What is the least common multiple of two or more numbers ? State the principles involved in finding it. Give the rule for finding it. What is the multiple when the numbers have no common prime fac- tors ? COMMON MULTIPLE. 141 factors of any one of the numbers, as 8, and such other prime fac- tors of the others, 6 and 12, as are not found among the prime fac- tors of 8 ; that is, the factor 3 : hence, 2 x 2 x 2 x 3 = 24, the least common multiple. To find the least common multiple of several numbers. I. Place the numbers on the same line, and divide by any- prime number that will exactly divide two or more of them, and set down in a line below the quotients and the undivided numbers. II. Then divide as before until there is no prime number greater than 1 that will exactly divide any two of the numbers. III. Then multiply together the divisors and the numbers of the lower line, and their product will be the least common multiple. NOTE. 1. The object of dividing by any prime number that will divide two or more of the numbers, is to find common factors. x 2. If the numbers have no common prime factor, their product will be their least common multiple. EXAMPLES. OPERATION. 1. Find the least common mul- tiple of 3, 4 and 8. 2)3 4 8 Ans. 2x2x3x1x2 = 24. 2)3 2. Find the least common mul- tiple of 3, 8 and 9. 3)3 8 9 Ans. 3x1x8x3=72. 1 8 3 3. Find the least common multiple of 6, 7, 8 and 10. 4. Find tKe least common multiple of 21 and 49. 5. Find the least common multiple of 2, 7, 5, 6, and 8. 6. Find the least common multiple of 4, 14, 28 and 98 7. Find the least common multiple of 13 and 6. 8. Find the least common multiple of 12, 4 and 7. 9. Find the least common multiple of 6, 9, 4, 14 and 16. 10. Find the least common multiple of 13, 12 and 4. 11. Find the least common multiple of 11, 17, 19, 21, and 14:2 CANCELLATION. CANCELLATION. 141. CANCELLATION is a method of shortening Arithmeti- cal operations by omitting or cancelling common factors. 1. Divide 24 by 12. First, 24 = 3 x 8 ; and 12 = 3 x 4. ANALYSIS. Twenty-four divided by 12 is OPERATION. equal to 3 x 8 divided by 3 x 4 ; by cancelling 24 $ x 8 or striking out the 3's, we have 8 divided by ~~nr ~* ~r = 2 4, which is equal to 2. 142. The operations in cancellation depend on two princi- ples : 1. The cancelling of a factor, in any number, is equivalent to dividing the number by that factor. 2. If the dividend and divisor be both divided by the same number, the quotient will not be changed. PRINCIPLES AND EXAMPLES. 1. Divide 63 by 21. ANALYSIS. Resolve tlie dividend and divi- OPERATION. sor into factors, and then cancel those which 63 _ * x 9 are common. " 2. In 7 times 56, how many times 8 ? ANALYSIS. Resolve 56 into the OPERATION. two factors 7 and 8, and then cancel 56x7_$x7x7 the 8. -g- -J- 3. In 9 times 84, how many times 12 ? 4. In 14 times 63, how many times 7 ? 5. In 24 times 9, how many times 8 ? 6. In 36 times 15, how many times 45 ? ANALYSIS. We see that 9 is a factor of 36 and 45. Divide by this factor, and write the OPERATK N. quotient 4 over 36, and the quotient 5 below 4 3 45. Again, 5 is a factor of 15 and 5. Divide $6 x I'SJ 15 by 5, and write the quotient 3 over 15. _ =1 Dividing 5 by 5, reduces the divisor to 1, which 40 need not be set down : hence, the true quotient $ 4x3=12. 141. What is cancellation ? 143. On what do the operations of rnneellitlon depend ? CANCELLATION. 143 143. Therefore, to perform the operations of cancellation : 1. Resolve the dividend and divisor into such factors as shall give all the factors common to both. II. Cancel the common factors and then divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor. NOTES. 1. Since every factor is cancelled by division, the quo- tient 1 always takes the place of the cancelled factor, but is omit- ted when it is a multiplier of other factors. 2. If one of the numbers contains a factor equal to the product of two or more factors of the other, they may all be cancelled. 3. If the product of two or more factors of the dividend is equal to the product of two or more factors of the divisor, such products may ba cancelled. 4- It is generally more convenient to set the dividend on the right of a vertical line and the divisor on the left. EXAMPLES. 1. What number is equal to 36 multiplied by 13 and the product divided by 4 times 9 ? ANALYSIS. We may place the numbers whose OPERATION. product forms the dividend on the right of a verti- ^ #0 cal line, and those which form the divisor on the A 10 left. We see that 4x9=36 ; we then cancel 4, 9, and 36. Ans. 13. 2. What is the result of 20 x 4 x 12, divided by 10x16x3? OPERATION. ANALYSIS. First, cancel the factor 10, in 10 and 20, and write the quotients 1 and 2 above the numbers. We then see that 16 x 3 48, and that 4x12=48; cancel 16 and 3 in the divisor, and 4 and 12 in the dividend ; hence, the quo- tient is 2. Am 3. Divide the product of 126 x 16 x 3, by 7 x 12. ANALYSIS We see that 7 is a factor OPERATION. of 126 giving a quotient of 18. We 1 cancsl 7, and place 18 at the right of 126. We then cancel 6, in 12 and 18, \ * and write the quotients 2 and 3. We ^ then cancal the factor 2, in 2 and 16, X and set down the quotients 1 and 8. Ans. 3x8x3 = ' The product of 1x1 is the divisor, and the product of 3 x 8 x 3 = 72, the dividend. 14:4: CANCELLATION. 4. What is the quotient of 3x8x9x7x15, divided by 63x24x3x5? ANALYSIS. The 63 is cancelled by 7 x 9 ; 24 by 3 x 8 ; 3 aiid 5, by 15 ; hence, the quotient is 1. OPERATION. $ H 5. Divide the product of 6x1x9x11, by 2x3x7x3 X21. 6. Divide the product of 4 X 14 x 16 x 24, by 7 x 8x32 Xl2. 7. Divide the product of 5 x 11 x 9 x 7 x 15 x 6, by 30 x 3 x21 x3x5. 8. Divide the product of 6 x 9 x 8 x 11 x 12 x 5, by 27 x 2 x 32 x 3. 9. Divide the product of 1 x 6 x 9 x 14 x 15 x 7 x 8, by 36 x 126x56x20. 10. Divide the product of 18 x 36 x 72 x 144, by 6 x 6 x 8 x 9x12x8. 11. Divide the product of 4 x 6 x 3 x 5, by 5 x 9 x 12 x 16. 12. Multiply 288 by 16, and divide the product by 8 x 9 x2x2. 13. In a certain operation the numbers 24, 28, 32, 49, 81, are to be multiplied together and the product divided by 8x4x7x9x6: what is the result ? 14. Multiply 240 by 18 and divide the product by 6 times 90. 15. Divide 16 x 20 x 8 x 3, by 30 x 8 x 6. 16. How many pounds of butter worth 15 cents a pound, may be bought for 25 pounds of tea at 48 cents a pound ? 1 7. How much calico at 25 cents a yard must be given for 100 yards of Irish sheeting at 87 cents a yard ? 18. How many yards of cloth at 46 cents a yard must be given for 23 bushels of rye at 92 cents a bushel ? 143. Give the rule for the operation of cancellation. CANCELLATION. 145 19. How many bushels of oats at 42 cents a bushel must be given for 3 boxes of raisins each containing 26 pounds, at 14 cents a pound ? 20. A man buys 2 pieces qf cotton cloth, each containing 33 yards at 11 cents a yard, and pays for it in butter at 18 cents a pound : how many pounds of butter did IIQ give ? 21. If sugar can be bought for 7 cents a pound, how many bushels of oats at 42 cents a bushel must I give for 56 pounds ? 22. If wool is worth 36 cents a pound, how many pounds must be given for 27 yards of broadcloth worth 4 dollars a yard? 23. If cotton cloth is worth 9 cents a yard, how much must be given for 3 tons of hay worth 15 dollars a ton ? 24. How much molasses at 42 cents a gallon must be given for 216 pounds of sugar at 7 cents a pound? 25. Bought 48 yards of cloth at 125 cents a yard : how many bushels of potatoes are required to pay for it at 150 cents a bushel ? 26. Mr. Butcher sold 342 pounds of beef at 6 cents a pound, and received his pay in molasses at 36 cents a gallon : how many gallons did he receive ? 27. Mr. Farmer sold 1263 pounds of wool at 5 cents a pound, and took his pay in cloth at 421 cents a yard : how many yards did he take ? 28. How many firkins of butter, each containing 56 pounds, at 18 cents a pound, must be given for 3 barrels of sugar, each containing 200 pounds, at 9 cents a pound ? 29. How many boxes of tea, each containing 24 pounds, worth 5 shillings a pound, must be given for 4 bins of wheat, each containing 145 bushels, at 12 shillings a bushel ? 30. A worked for B 8 days, at 6 shillings a day, for which he received 12 bushels of corn : how much was the corn worth a bushel ? 31. Bought 15 barrels of apples, each containing 2 bushels at the rate of 3 shillings a bushel : how many cheeses, each weighing 30 pounds, at 1 shilling a pound, will pay for the apples ? 10 14:6 COMMON FRACTIONS. COMMON FRACTIONS. 144. The unit 1 denotes an entire thing, as 1 apple, 1 chair, 1 pound of tea. If the unit 1 be divided into two equal parts, each part is called one-half. If the unit 1 be divided into three equal parts, each part is called one-third. If the unit 1 be divided into four equal parts, each part is called one-fourth. If the unit 1 be divided into twelve equal parts, each part is called one-twelfth ; and if it be divided into any number of equal parts, we have a like expression for each part. The parts are thus written : is read, one-half. -f is read, one-seventh, one-third | - - one-eighth, one-fourth. . T\T ~ - one-tenth. - one-fifth. T ^ - - one-fifteenth, one-sixth. ^ - - one-fiftieth. The i, is an entire half; the J, an entire third ; the J, an entire fourth ; and the same for each of the other equal parts : hence, each equal part is an entire thing, and is called a frac- tional unit. The unit 1 , or whole thing which is divided, is called the unit of the fraction. NOTE. In every fraction let the pupil distinguish carefully between the unit of the fraction and the fractional unit. The first is the whole thing from which the fraction is derived ; the second, one of the equal parts into which that thing is divided. 145. Each fractional unit may become the base of a col- lection of fractional units : thus, suppose it were required to express 2 of each of the fractional units : we should then write 144. What is a unit ? What is each part called when the unit 1 is divided into two equal parts ? When it is divided into 3 ? Into 4? Into 5? Into 12? How may the one-half be regarded ? The one-third ? The one-fourth ? What is each part called ? What is the unit of a fraction ? What is a fractional unit ? How do you distinguish between the one and the otlu-r ? COMMON FRACTIONS. which is read 2 halves = J x 2 " " " 2 thirds =Jx2 2fourths=Jx2 2 fifths =x2 &c., &c., &c. f &c. If it were required to express 3 of each of the fractional units, we should write % -| which is read 3 halves =^ x 3 f " 3 thirds =4x3 " " " 3 fourths =1x3 J " " " 3 fifths =1x3 &c., &c., &c., &c. ; hence, A FRACTION is one of the equal parts of the unit 1, or a collection of such equal parts. Fractions are expressed by two numbers, the one written above the other, with a line between them. The lower num- ber is called the denominator, and the upper number the numerator. The denominator denotes the number of equal parts into which the unit is divided ; and hence, determines the value of the fractional unit. Thus, if the denominator is 2, the fractional unit is one-half; if it is 3, the fractional unit is one- third ; if it is 4, the fractional unit is one-fourth, &c., &c. The numerator denotes the number of fractional units taken. Thus, -f denotes that the fractional unit is ^, and that 3 such units are taken ; and similarly for other fractions. In the fraction f , the base of the collection of fractional units is , but this is not the primary base. For, is one- fifth of the unit 1 ; hence, the primary base of every fraction is the unit 1. 145. May a fractional unit become the base of a collection ? What is a fraction ? How are fractions expressed ? What is the lower number called ? What is the upper number called ? What does the denomina- tor denote? What does the numerator denote? In the fraction 3 fifths, what is the fractional base ? What is the primary base ? What is the primary base of every fraction ? 148 COMMON FRACTIONS. 146. If we take other units 1, each of the same kind, and divide each into equal parts, such parts may be expressed in the same collection with the parts of the first : thus, f is read 3 halves. I " " ? fourths. i/- " " 16 fifths. V " . *' 18 sixths. 2j&- 25 sevenths. 147. A whole number may be expressed fractionally by writing 1 below it for a denominator. Thus, 3 may be written -f- and is read, 3 ones. 5-- - {--- 5 ones. 6 - - f - - - 6 ones. 8 - - - -f- - - - 8 ones. But 3 ones are equal to 3, 5 ones to 5, 6 ones to 6, and 8 ones to 8 ; hence, the value of a number is not changed by placing 1 under it for a denominator. 148. If the numerator of a fraction be divided by its de- nominator, the integral part of the quotient will express the number of entire units used in forming the fraction ; and the remainder will show how many fractional units are over. Tims, JyL are equal to 3 and 2 thirds, and is written -V- 3 I : hence, A fraction has the same form as an unexecuted division. From what has been said, we conclude that, 1st. A fraction is one or more of the equal parts of the unit 1. 2d. The denominator shows into how many equal parts the unit is divided, and hence indicates the value of the fractional unit : 146. If a second unit be divided into equal parts, may the parts be expressed with those of the first? How many units have been divided to obtain 6 thirds ? To obtain 9 halves ? 12 fourths ? 147. How may a whole number be expressed fractionally? Does this change the value of the number? 148. If the numerator be divided by the denominator, what docs the quotient show? What does the remainder show? What form has a fraction ? What are the seven principles which follow ? COMMON FRACTIONS. 149 3d. The numerator shows how many fractional units are taken : 4th. The value of every fraction is equal to the quotient arising from dividing the numerator by the denominator. 5th. When the numerator is less than the denominator, the value of the fraction is less than 1. 6th. When the numerator is equal to the denominator, the value of the fraction is equal to 1. 7th. When the numerator is greater than the denomina- tor, the value of the fraction is greater than 1 EXAMPLES IN WRITING AND READING FRACTIONS. 1. Read the following fractions ; T 5 u, f , , T 7 o, f , 5 9 o, TT. What is the unit of the fraction, and what the fractional unit, in each example ? How many fractional units are taken in each? 2. Write 12 of the 17 equal parts of 1. 3. If the unit of the fraction is 1, and the fractional unit one-twentieth, express 6 fractional units. Express 12, 18, 16, 30, fractional units. 4. If the fractional unit is one 36th, express 32 fractional units ; also, 35, 38, 54, 6, 8. 5. If the fractional unit is one-fortieth, express 9 fractional units ; also, 16, 25, 69, 75. DEFINITIONS. 149. A PROPER FRACTION is one whose numerator is less than the denominator. Tue following are proper fractions : i i i I f J, A, t, * 150. An IMPROPER FRACTION is one whose numerator is equal to, or exceeds the denominator. NOTE. Such a . fraction is called improper because its value equals or exceeds 1. 149. What is a proper fraction ? Give examples. 150. What is an improper fraction ? Why improper ? Give exam- ples. 150 PROPOSITIONS IN The following are improper fractions : 4, 4, 4, 4, f , 4, , , V- 151. A SIMPLE FRACTION is one whose numerator and de- nominator are both whole numbers. NOTE. A simple fraction may be either proper or improper. The following are simple fractions : i f , *, f , 4, 4, 4. * 152. A COMPOUND FRACTION is a fraction of a fraction, or several fractions connected by the word of, or x . The following are compound fractions : Jofi iofiofj, x3, ixJx-4. 153. A MIXED NUMBER is made up of a whole number and a fraction. The following are mixed numbers : 3i, 41, 6f, 54, 6|, 3f 154. A COMPLEX FRACTION is one whose numerator or de- nominator is fractional ; or, in which both are fractional. The following are complex fractions : j 2 f 45t 5 191' *' 69V 155. The numerator and .denominator of a fraction, taken together, are called the terms of the fraction : hence, every fraction has two terms. FUNDAMENTAL PROPOSITIONS. 156. By multiplying the unit 1, we form all the whole numbers, 151. What is a simple fraction ? Give examples. May it be proper or improper ? 153. What is a compound fraction ? Give examples. 153. What is a mixed number ? Give examples. 154. What is a complex fraction ? Give examples. 155. How many terms has every fraction ? What are they ? 156. How may all the whole numbers be formed? How may the fractional units be formed ? How many times is one-half less than 1 ? How many times is any fractional unit less than 1 ? COMMON FRACTIONS. 151 2, 3, 4, 5, 6, 1, 8, 9, 10, &c. ; and by dividing the unit 1 by these numbers we form all the fractional units, i' 4' I* i> i' I' i> I' A &c - Now, since in 1 unit there are 2 halves, 3 thirds, 4 fourths, 5 fifths, 6 sixths, &c., it follows that the fractional unit becomes less as the denominators are increased : hence, The fractional unit is such a part of I, as I is of the denominator of the fraction. Thus, J is such a part of 1, as 1 is of 2 ; J is such a part of 1, as 1 is of 3-; J is such a part of 1 as 1 is of 4, &c. &c. 157. Let it be required to multiply by 3. ANALYSIS. In f there are 5 fractional OPERATION-. units, each of which is ^, and these are to 4 x 3^-5-vp- J^A be taken 3 times. But 5 things taken 3 times, gives 15 things of the same kind ; that is, 15 sixths : hence, the product is 3 times as great as the multiplicand : therefore, we have PROPOSITION I. If the numerator of a fraction be multi- plied by any number, the value of the fraction will be in- creased as many times as there are units in the multiplier. 4. Multiply T V by 14. 5. Multiply % by 20. 6. Multiply Jj&z- by 25 EXAMPLES. 1. Multiply -3 by 8. 2. Multiply I by 5. 3. Multiply \ by 9. 158. Let it be required to multiply by 3. ANALYSIS. In there are 4 fractional OPERATION. units, each of which is . If we divide 4- X 3 4 . the denominator by 3, we change the frac- 6 ~ 3 tional unit to \, which is 3 times as great as , since the first is contained in 1, 2 times, and the second 6 times. If we take this fractional unit 4 times, the result , is 3 times as great as $: therefore, we have PROPOSITION II. If the denominator of a fraction be divi- ded by any number, the value of the fraction will be in- creased as many times as there are units in that number. 157. What is proved in Proposition I. ? 152 PROPOSITIONS IN EXAMPLES. 4. Multiply H by 2, 4, 6. 5. Multiply by 2, 6, 7. 6. Multiply $fo by 5, 10. 1. Multiply | by 2, by 4. 2. Multiply Jf by 2, 4, 8. 3. Multiply ^ by 2, 4, 6. 159. Let it be required to divide fa by 3. ANALYSIS. In -ft, there are 9 fractional OPERATION. units, each of which is -, 1 ,-, and these are s ' -f-3 9-3 -3 to be divided by 3. But 9 things, divided 1 1 by 3, gives 3 things of the same kind for a quotient ; hence, the quotient is 3 elevenths, a number one-third as great as -ft ; hence, we have PROPOSITION III. If the numerator of a fraction be divi- ded by any number, the value of the fraction will be dimin- ished as many times as there are units in the divisor. EXAMPLES. 1. Divide ff by 2, by 7 2. Divide $J by 56. 3. Divide f by 25, by 8. 4. Divide ff by 8, 16, 10. 1GO. Let it be required to divide fa by 3. ANALYSIS. In -ft-, there are 9 fractional OPERATION. units, each of which is -ft-. Now. if we $ -^-3=^ *-r. multiply the denominator by 3 it becomes 33, and the fractional unit becomes -^-j, which is only ^ of -, 1 ,-, be- cause 33 is 3 times as great as 11. If we take this fractional unit 9 times, the result, -,-, is exactly ^ of -ft : hence, we have PROPOSITION IY. If the denominator of a fraction be multiplied by any number, the value of the fraction will be diminished as many times as there are units in that number. EXAMPLES. 1. Divide \ by 2. 2. Divide by 1. 3. Divide -^ by 4. 4. Divide f by 8. 5. Divide fj- by 17. 6. Divide T V% by 45. 158. What is proved in proposition II. ? 159. What is proved in proposition III. ? 100. What is proved in proposition IV. ? COMMON FRACTIONS. 153 161. Let it be required to multiply both terms of the frac- tion f by 4. ANALYSIS. In f, the fractional unit is , and it OPERATION. is taken 3 times. By multiplying the denominator ?lf -JL2.. by 4, the fractional unit becomes ^7, the value of 5x4~~^o which is ^ times as as great as i. By multiplying the numerator by 4, we increase the number of fractional units taken, 4 times, that is, we increase the number just as many times as we decrease the value ; hence, the value of the fraction is not changed ; there- fore, we have PROPOSITION Y. If both terms of a fraction be multiplied by the same number, the value of the fraction will not be changed. EXAMPLES. 1. Multiply the numerator and denominator of -f- by 7 : this gires ^HM. 7 X 7 49 2. Multiply the numerator and denominator of -fa by 3, by 4, by 5, by 6, by 9. 3. Multiply each term of | by 2, by 3, by 4, by 5, by 6. 162. Let it be required to divide the numerator and de- nominator of T 6 3- by 3. ANALYSIS. In -rV, the fractional unit is -fa, and OPERATION. is taken 6 times. By dividing the denominator 6 -r-3__2 by 3, the fractional unit becomes i, the value of T^_^o 7"* which is 3 times as great as -fa. By dividing the numerator by 3, we diminish the number of fractional units taken 3 times : that is, we diminish the number just as many times as we increase the value : hence, the value of the fraction is not changed : therefore we have PROPOSITION YI. If both terms of a fraction be divided by the same number, the value of the fraction will not be changed. EXAMPLES. 1. Divide both terms of the fraction ^ by 2 : this gives = Ans. 161. What is proved hy proposition V. ? 162. What is proved by proposition VI. ? 154 REDUCTION OF 2. Divide both terms by 8 : this gives ^ f = J. 3. Divide both terms of the fraction -j 3 ^- by 2, by 4, by 8, by 16. 4. Divide both terms of the fraction T ^j by 2, by 3, by 4, by 5, by 6, by 10, by 12. REDUCTION OF FRACTIONS. 163. REDUCTION OF FRACTIONS is the operation of changing the fractional unit without altering the value of the fraction. A fraction is in its lowest terms, when the numerator and denominator have no common factor. CASE i. 164. To reduce a fraction to its lowest terms. 1. Reduce T W to its lowest terms. ANALYSIS. By inspection, it is seen that 5 is a common factor of the numerator and IST OPERATION. denominator. Dividing by it, we have if. 5) T 7 -*y r 4-i. We then see that 7 is a common factor of 14 and 35: dividing by it, we have . Now, fr\i 4 _ 2 there is no common factor to 2 and 5 : hence, 'So t' is in its lowest terms. The greatest common divisor of 70 and 175 2D OPERATION. is 35, (Art. 136); if we divide both terms of 35) TJ^ .2. the fraction by it, we obtain . The value of the fraction is not changed in either operation, since the numera- tor and denominator are both divided by the same number (Art. 162): hence, the following RULE. Divide the numerator mid denominator by any number that will divide them both without a remainder, and divide the quotient, in the same manner until they have no common factor. Or : Divide the numerator and denominator by their great- est common divisor. 163. What is reduction of fractions ? When is a fraction in its lowest terms ? 164. How do you reduce a fraction to its lowest terms ? COMMON FRACTIONS. 155 EXAMPLES. Reduce the following fractions to their lowest terms. 1. Reduce -ff. 2. Reduce ff. 3. Reduce f. 4. Reduce 5. Reduce 6. Reduce V. Reduce 8. Reduce 9. Reduce 10. Reduce 11. Reduce 12. Reduce 13. Reduce 14. Reduce 15. Reduce 16. Reduce CASE II. 165. To reduce an improper fraction to its eouivalent whole or mixed number. 1. In $/ how many entire units ? ANALYSIS. Since there are 8 eighths in 1 unit, OPERATION. in * there are as many units as 8 is contain- 8)59 ed times in 59, which is 7| times. =-- Hence, the following RULE. Divide the numerator by the denominator, and the result ivill be the whole or mixed number. EXAMPLES. 1. Reduce & and fy to their equivalent whole or mixed numbers. OPERATION. OPERATION. 4)84 9)67 2. Reduce sg. to a whole or mixed number. 3. In I? 9 - yards of cloth, how many yards ? 4. In -^L of bushels, how many bushels ? 165. How do you reduce an improper fraction to a whole or mixed number ? 156 REDUCTION OF 5. If I give I of an apple to each one of 15 children, how many apples do I give ? 6. Reduce ffj, 3ff, JtfffiL, *fj#f. t to their whole or mixed numbers. 7. If I distribute 878 quarter-apples among a number of boys, how many whole apples do I use ? 8. Reduce % 5 T 8 ^, \W, WsWeS to tneir whole or mixed numbers. 9. Reduce JLt^ffi^ J^\^a, 2^p } to t h e i r w hole or mixed numbers. CASE III. 160. To reduce a mixed number to its equivalent improper fraction. 1. Reduce 4f- to its equivalent improper fraction. ANALYSis.-Since ' in any number OPERATION. there are 5 times as many fifths as A .. r O n Gp^ n units, in 4 there will be 5 times 4 fifths, or 20 fifths, to which add 4 fifths, and add 4 fifths. we have 24 fifths. gives %* = 24 fifths. Hence, the following RULE. Multiply the whole number by the denominator of the fraction : to the product add the numerator, and place the sum over the given denominator. EXAMPLES. 1. Reduce 47f to its equivalent fraction. 2. In It yards, how many eighths of a yard? 3. In 42 -/^ rods, how many twentieths of a rod ? 4. Reduce 625-^- to an improper fraction. 5. How many 112ths in 205 T 4 T % ? 6. In 84^ days, how many twenty-fourths of a day ? 7. In 15J$| years, how many 365ths of a year ? 8. Reduce 916-{} to an improper fraction. 9. Reduce 25 T %-, 156f^, to their equivalent fractions. 100. How do you reduce a mixed number to its equivalent improper fraction. COMMON FRACTIONS. 157 CASE IV. 167. To reduce a whole number to a fraction having a given denominator. 1. Reduce 6 to a fraction whose denominator shall be 4. ANALYSIS. Since in 1 unit there are 4 fourths, OPERATION. it follows that in 6 units there are 6 times 4 fourths, 6x4 24. or 24 fourths: therefore, 6=Y hence, .gjt RULE. Multiply the whole number and denominator together, and write the product over the required denomi- nator. EXAMPLES. 1. Reduce 12 to a fraction whose denominator shall be 9. 2 Reduce 46 to a fraction whose denominator shall be 15. 3. Change 26 to 7ths. 4. Change 178 to 40ths. 5. Reduce 240 to IHths. 6. Change $54 to quarters. 7. Change 96?/<^. to quarters. 8. Change 426/6. to 16ths. CASE V. 168. To reduce a compound fraction to a simple one. 1. What is the value of of f? ANALYSIS. Three-fourths of f is 3 times 1 fourth OPERATION. of $ ; 1 fourth of f is & (Art. 160) ; 3 fourths of f is 3x5 15 3 times &, or if : therefore, f of $=i : hence, -= = 4x7 2o RULE. Multiply the numerators together for a new numerator, and the denominators together for a new de- nominator. NOTE. If there are mixed numbers, reduce them to their equiv- alent improper fractions. EXAMPLES. P*educe the following fractions to simple ones. 1. Reduce J of J of f. 2. Reduce of of f. 3. Reduce f of f o 4. Reduce 2J of 6J of 7. 5. Reduce 5 of \ of | of 6. 6. Reduce 6^ of 7} of 6ff. 158 REDUCTION OF METHOD BY CANCELLING. 169. The work may often be abridged by cancelling com- mon factors in the numerator and denominator (Art. 143). In every operation in fractions, let this be done whenever it is possible. EXAMPLES. 1. Reduce f of f of -f to a simple fraction. 5 Here, 7 | 5=f NOTE. The divisors are always written on the left of the vertical line, and the dividends on the right. 2 2. Reduce of f of T ^ to its simplest terms. ! * * 2 * * -rr V V r ^ TT ^-o __- NX V ^^ T- . rv-t* xi ere, i A A x vet F; UI R 5 | 2=2. NOTE. Besides cancelling the like factors 8 and 8, and 9 and 9> we also cancel the factor 3, common to 15 and 6, and write ovei them, and at the left and right, the quotients 5 and 2. 3. Reduce | of -f of of -fife of T 5 ^ to its simplest terms. 4. Reduce -f-fc of T \ of T % of f to its simplest terms. 5. Reduce 3|- of f of ^ of 49 to its simplest terms. CASE TI. 170. To reduce fractions of different denominators to fractions having a common denominator. 1. Reduce \, % and 4 to a common denominator. 167. How do you reduce a whole number to a fraction having a given denominator? 168. How do you reduce a compound fraction to a simple one ? 169. How is the reduction of compound fractions to simple ones abridged by cancellation. COMMON FRACTIONS. 159 ANALYSIS. If both terms of the OPERATION. first fraction be multiplied by 15, 1x3x5=15 1st num. the product of the other denomina- 7x2x5 = 70 2d num. tors, it will become ft. If both i v 3v9 24- 3r1 nnm terms of the second fraction be mul- tiplied by 10, the product of the 2x3x5 = dO clenom. other denominators, it will become $. If both terms of the third be multiplied by 6, the product of the other denominators, it will become f . In each case, we have multiplied both terms of the fraction by the same number ; hence, the value has not been altered (Art. 161) : hence, the following RULE. Eeduce to simple fractions when necessary ; then multiply the numerator of each fraction by all the denomi- nators except its own, for the new numerators, and all the denominators together for a common denominator. NOTE. When the numbers are small the work may be per- formed mentally. Thus, i- \f *= EXAMPLES. Reduce the following fractions to common denominators. 1. Reduce f, f, and -$-. 2. Reduce f , -f^-, and f . 3. Reduce 4f-, |, and $. 4. Reduce 2J, and J of -f. 5. Reduce 5 J,f of J, and 4. 6. Reduce 3 of J and f . 7. Reduce ,Y/, and 37. 8. Reduce 4, fj, and . 9. Reduce 7J, ffr, 6J. 10. Reduce 4, 8|, and 2|. NOTE. We may often shorten the work by multiplying the nu- merator and denominator of each fraction by such a number as will make the denominators the same in all. 10. Reduce J and J to a common denominator. OPERATION. ANALYSIS. Multiply both terms of the first by 1=4 3, and both terms of the second by 2. ls. 3 < 11. Reduce and J. 12. Reduce , ^, and }. 13. Reduce -. 14. Reduce f , 3, and |. 15. Reduce 6^, 9J,and5. 16. Reduce 7f,f, J, and. 170. How do you reduce fractions of different denominators to frac- tions having a common denominator ? When the numbers are small, how may the work be performed ? 160 REDUCTION OF CASE VII. 171. To reduce fractions to their least common denominator. The least common denominator is the number which con- tains only the prime factors of the denominators. 1. Reduce J, f , and |, to their least common denominator. OPERATION. (12-=-3)xl = 4 1st Numerator. 3)3 . 6 . 4 (12-^-6) x 5 = 10 2d " 2)1 . 2 .~4~ (12-T-4)x3= 9 3d " 1.1.2 3x2x2 = 1 2, least com. denom. Therefore, the fractions J, f, and f, reduced to their least common denominator, are T %, -ff, and T \. Hence, the following RULE, I. Find the least common multiple of the denomi- nators (Art. 140), which will be the least common denominator of the fractions. II. Divide the least common denominator by the denomina- tors of the given fractions separately, and multiply the nume- rators by the corresponding quotients, and place the products over the least common denominator. NOTES. 1. Before beginning the operation, reduce every frac- tion to a simple fraction and to its lowest terms. 2. The expressions, (12-r-3)xl, (12-7-6) x 5, (12-f-4)x3, indi- cate that the quotients are to be multiplied by 1, 5, and 3. EXAMPLES. Reduce the following fractions to their least common denominator. 2. Reduce f , f , T 3 T . 3. Reduce 14f, 6-f, 5J. 4. Reduce -^ -fa, f . 5. Reduce -flfr, ^, f. 6. Reduce , 3^, 4. 1. Reduce 3|, 8. Reduce J, , j, and . 9. Reduce 2J of , 3} of 2. 10. Reduce -f, f , , and T V 11. Reduce J, f, f, I . 171. Wliat is the least common denominator of several fractions? How do you reduce fractions to their least common denominator V COMMON FRACTIONS. 161 OPERATION. OPERATION. ADDITION OF FRACTIONS. 172. Addition of Fractions is the operation of finding the number of fractional units in two or more fractions. 1. What is the sum of J, f , and f ? ANALYSIS. The fractional unit is the same in each fraction, viz. : ^ ; but the numerators show how many such units are taken (Art. 148) ; hence, the sum of the numerators written over tJie common denominator, expresses the sum of Ans. f =4. the fractions. 2. What is the sum of J and f ? ANALYSIS. In the first, the fractional unit is , in the second it is ^. These unite, not being of the same kind, cannot be expressed in the same collection. But the =f, and f =$, in each of which the unit is : hence, their sum is ^=1^. NOTE. Only units of the same kind, whether fractional or inte- gral, can be expressed in the same collection, From the above analysis, we have the following RULE. I. When the fractions have the same denominator, add the numerators, and place the sum over the common deno- minator. II. When they have not the same denominator, reduce them to a common denominator, and then add as before. NOTE. After the addition is performed, reduce every result to its lowest terms. *-* EXAMPLES. 1. Add J, f , f , and f . 2. Add |, f , and f 3. Addf, f,^,an 4. Add t^,^, an 5. Add f , .ft, and ft. 6. Add i, |, f, and ft. 7. Add |, I fc and ft. 8. Add |, I i and -ft. 9. Add 9, |, T V, f , and f . 10. Add J, f , f , 1, and f 11. Add f V, f , A, and f . 12. Add |, f , and f. 13. Add T V, f , f, and f . 14. Add -!%, f, f, and ^. 162 SUBTRACTION OF 15. What is the sum of 19}, 6, and 4|? OPERATION. Whole numbers. Fractions. 19 + 6+4=29^ ^ *++*=*= 17o. NOTE. When there are mixed numbers, add the uhole, numbers and fractions separately, and then add their sums. Find the sums of the following fractions : 16. Add 3J, 7y%, 12f, 1?. 20. Add 900 T V, 450, 17. Add 16, 9|, 25, T . 21. AddJof T 3 T of T to 18. Add | of |, 4. of 9, 14 T V 22. Add 17| to f of 27$. 19. Add 2 T 8 T , 6, and 12-if. 23. Add $, 7J, and 8|. 24. What is the sum cf | of 12 of 7|, and $ of 25 ? 25. What is the sum of -fa of 9f and -^ of 328f ? 174. 1. What is the sum of -J- and ? NOTE. If each of the two fractions has OPERATION. 1 for a numerator, the sum of the frac- A- +1 c + 5 il tions will be equal to the sum of their _5 + G _ denominators divided by their product. ~5 j^~ G " ao' 2. What is the sum of | and ^- ? of and T V ? 3. What is the sum of -f and -fa ? of T \j- and y 1 ^- ? of T ^ andi? 4. What is the sum of J and yV? f 1 and ? of J and yV ? SUBTRACTION OF FRACTIONS. 175. SUBTRACTION of Fractions is the operation of finding the difference between two fractions. 173. What is addition of fractions ? When the fractional unit is the same, what is the sum of the fractions ? What units may be expressed in the same collection ? What is the rule for the addition of fractions ? 173. When there are mixed numbers, how do you add ? 174. When two fractions have 1 for a numerator, what is their sum equal to ? 175. What is subtraction of fractions ? COMMON FBACTIONS. 163 1. What is the difference between and f ? ANALYSIS. In this example the fractional unit is i : there are 5 such units in the minuend and 3 in the subtrahend : their difference is 2 eighths ; therefore, 2 is written over the common denomi- nator 8. 2. From J^. take -i 3. From -| take f . 4. From 5. From OPERATION. take take OPERATION. i . 4 jj, __ y* _ g _ ttr ~T1F TT 6. What is the difference between and ANALYSIS. Reduce both to the same frac- tional unit -^ : then, there are 10 sucli units in the minuend and 4 in the subtrahend: hence, the difference is 6 twelfths. From the above analysis we have the following RULE. I. When the fractions have the same denominator, subtract the less numerator from the greater, and place the difference over the common denominator. II. When they have not the same denominator, reduce them lo a common denominator, and then subtract as before. EXAMPLES. Make the following subtractions : 1 . From -f- take f. 2. From f take f. 3. From - take - 4. From 1, take -fifo. 5. From of 12, take ff of J. 6. F'mf of 1J of 7, take j. off. 7. From f of J of J take -ft of of 1. 8. From of J of 6J, take f of f of f . 9. From T * T of f of J, take ^ of ^. 10. What is the difference between 41 and OPERATION. or > 16i MULTIPLICATION OF 176. Therefore : When there are mixed numbers, change both to improper fractions and subtract as in Art. 11.5 ; or, subtract the integral and fractional numbers separately, and write the results. 11. From S4-& take 16J. | 12. From 246f take 164. 13. From 7 take 4} : ^ =1 ft. and 1=^. NOTE. Since we cannot take & from -/,- we OPERATION. borrow 1, or ||, from the minuend, which added 7*=7T&- to ^r=H J then f f from f leaves f<f. We must 41 4 V now carry 1 to the next figure of the subtrahend and proceed as in subtraction of simple numbers. Ans. 2|-^ 14. From 16* take 5f 16. From 36f take 27^. 15. From 26f take 19f It. From 400 T \ take 327*. 18. From J take ^. NOTE. When the numerators are 1, OPERATION. the difference of the two fractions is l_ T i_ = l-_. equal to the difference of the denomina- i __ i _ tors divided by their product 19. What is the difference between ^ and J ? Between iand T V? ^and^V? A and - MULTIPLICATION OF FRACTIONS. 177. MULTIPLICATION of Fractions is the operation of taking one number as many times as there are units in another, when one of the numbers is fractional, or when they are both fractional. 1. If one yard of cloth cost of a dollar, what will 4 yards cost? ANALYSIS. Four yards will cost 4 OPERATION. times as much as 1 yard; if 1 yard J x4r=A ^ 2J costs 5 eighths of a dollar, 4 yards will cost 4 times 5 eighths of a dollar, which are 20 eighths ; equal to 2i dollars. 176. When there are mixed numbers, how do you subtract? Explain the case when the fractional part of the subtrahend is the greater ? 177. What is multiplication of fractions ? COMMON FKACTIOHS. 1G5 OPERATION. W /\ ^ tj j - 2d. If we divide the denominator by 4, OR, the fraction will be multiplied by 4 (Prop. o II) : performing the operation, we obtain, which 2i : hence, To multiply a fraction by a whole number : Multiply the numerator, or divide the denominator by the multiplier. EXAMPLES. 1. Multiply -^ by 12. 2. Multiply | by 7. 3. Multiply iff. by 9. 4. Multiply 1 T ~JL by 5. 5. Multiply -J-ff by 49. 6. Multiply i^f by 26. 7. If 1 dollar will buy f of a cord of wood, how much will 15 dollars buy ? 8. At | of a dollar a pound, what will 12 pounds of tea cost ? 9. If a horse cats J of a bushel of oats in a day, how much will 18 horses eat ? 10. What will 64 pounds of cheese cost, at -^ of a dollar a pound ? 11. If a man travel 2 of a mile an hour, how far will he travel in 16 hours? 12. At f of a cent a pound, what will 45 pounds of chalk cost? 13. If a man receive -^ of a dollar for 1 day's labor, how much will he receive for 15 days ? 14. If a family consume ^ of a barrel of flour in 1 month, how much will they consume in 9 months ? 15. If a person pays -j-J- of a dollar a month for tobacco, how much does he pay in 1 8 months ? 181. To multiply a whole number by a fraction. 1. At 15 dollars a ton, what will |- of a ton of hay cost? ANALYSIS. 1st. Four fifths of a ton will cost 4 times as much as 1 fifth of a ton ; if OPERATION. 1 ton cost 15 dollars, 1 fifth will cost i of 15 (15-i-5)x4 = 12 dollars, or 3 dollars, and i will cost 4 times 3 v dollars, which are 12 dollars. 180. How do you multiply a fraction by a whole number ? 166 MULTIPLICATION OF OR : 2d. 4 fifths of a ton will cost 1 fifth of 4 times the cost of 1 ton ; 4 times 15 is 60, 1 v l Z 10 and 1 fifth of 60 is 12. 4 NOTE. Both operations may be combined -"* 2 in one by the use of the vertical line and can- cellation : hence, | 12 Ans. Divide the whole number by the denominator of the fraction and multiply the quotient by the numerator ; Or : Multiply the whole number by the numerator of the fraction and divide the product by the denominator. EXAMPLES. 1. Multiply 24 by ?. 2. Multiply 42 by 3. Multiply 105 by 4. Multiply 64 by 5. What is the cost of of a yard of cloth at 8 dollars a yard ? 6. If an acre of land is valued at 75 dollars, what is -^ of it worth ? 7. If a house is worth 320 dollars, what is T 9 ^- of it worth ? 8. If a man travel 46 miles in a day, how far does he travel in of a day ? 9. At 18 dollars a ton, what is the cost of ^ of a ton of hay? 10. If a man earn 480 dollars in a year, how much does he earn in -J-J of a year ? 182. To multiply one fraction by another. 1 . If a bushel of corn cost f of a dollar, what will -f of a bushel cost ? OPERATION. ANALYSIS. 5-sixths of a bushel will cost Jx|$. ^. _ times as much as 1 bushel, or 5 times 4 ! 1 sixth of a bushel : i of is &, (Art. 180), g and 5 times -fa is $=$ : hence, 8 5 = 181. How do you multiply a whole number by a fraction ? COMMON FRACTIONS. 167 Multiply the numerators together for a new numerator and the denominators together for a new denominator. NOTES. 1. When the multiplier is less than 1, we do not take the whole of the multiplicand, but only such a part of it as the multiplier is of 1. 2. When the multiplier is a proper fraction, multiplication does not imply increase, as in the multiplication of Avhole numbers. The product is the same part of the multiplicand which the multi- plier is of 1. EXAMPLES. 1. Multiply I by 2. Multiply A by 3. Find the pro't of 4. Find the pro't of f ft, f \. |, J, 5. If silk is worth ft of a dollar a yard, what is f of a yard worth ? 6. If I own ^ of a farm and sell | of my share, what part of the whole farm do I sell ? 7. At of a dollar a pound, what will ft of a pound of tea cost ? 8. If a knife cost * of a dollar and a slate -f as much, what does the slate cost ? OPERATION. 9. Multiply 5 J by -J- of |. 5^=^ ; i O f f =- 21 v 8 - 7 NOTE. Before multiplying, * 3ir I ; reduce both fractions to the form * i of simple fractions. 9 | 1=1 Ans. GENERAL EXAMPLES. 1 . Mult, l of I of 4- by 2. Mult i by $ of If. 3. Mult. J of 3 by i of 4. Mult. 5 of | of f by 4J. 5. Mult. 14 of'-f of 9 by Gf 6. Mult, f of 6 of -| by f of 4. 183. When the multiplicand is a whole and the multi- plier a mixed number. 183. How do you multiply one fraction by another? When the multiplier is less than 1, what part of the multiplicand is taken ? If the fraction is proper, does multiplication imply increase ? What part is the product of the multiplicand ? 168 DIVISION OF 7. What is the product of 48 by 8 ? NOTE. First multiply 48 by , which gives 48 x = 8 8 ; then by 8, which gives 384, and the sum, 392 40 v Q OQJ. is the product : hence, 392 Multiply first by the fraction, and then by the whole number, and add the products. 8. Mult. 67 by 9, 9. Mult. 12 by 10. Mult. 108 by 1 11. Mult. 5f by 3|. 12. What is the product of 6|, 2 and J of 12 ? 13. What will 24 yards of cloth cost at 3| dollars a yard ? 14. What will 6 bushels of wheat cost at 3j dollars a bushel ? 15. A horse eats ^\ of - of 12 tons of hay in three months ; how much did he consume ? 16. Jf of of a dollar buy a bushel of corn, what will ^ of T 6 T of a bushel cost ? 17. What is the cost of 5| gallons of molasses at 96 J cents a gallon ? 18. What will 7| dozen caudles cost at T 3 T of a dollar per dozen ? 19. What must be paid for 175 barrels of flour at 7| dol- lars a barrel ? 20. If | of -f- of 2 yards of cloth can be bought for one dol- lar, how much can be bought for | of 13| dollars ? 21. What is the cost of 15| cords of wooc^at 3|- dollars a cord? DIVISION OF FRACTIONS. 184. Division of Fractions is the operation of finding a number which multiplied by the divisor will produce the divi- dend, when one or both of the parts are fractional. 185. To divide a fraction by a ivhole number. 1. If 4 bushels of apples cost -jj- of a dollar, what will 1 bushel cost ? 183. How may you, multiply when the multiplicand is a icJiolc and the multiplier a mixed number? 184. What is division of fractions? 185. How do you divide a fraction by a whole number ? COMMON FRACTIONS. 169 ANALYSIS. Since 4 bushels cost f of a dollar, J. bushel will cost \ of f of a dollar. Dividing the numerator of the fraction f by 4, we have (Art. 159). OPERATION. Multiplying the denominator by 4 will pro- A -^-4 -^j~~- duce the same result (Art. 160) : hence, Divide the numerator or multiply the denominator by the divisor. NOTE. By the use of the vertical line and the principles of cancellation (Art. 148), all operations in divisions of fractions may be greatly abridged. 9 | 2=f EXAM 1. Divide ff- by 6. 3. Divide ^f- by 9. 3. Divide ^ by 15. 4. Divide -fff by 75. PLES. 5. Divide || by 6. 6. Divide by 12. 7. Divide if by 20. 8. Divide iff by 27. 9. If 6 horses eat T ^j of a ton of hay in 1 month, how much will one horse eat ? 10. If 9 yards of ribbon cost f of a dollar, what will 1 yard cost? 11. If 1 yard of cloth cost 4 dollars, how much can be bought for f of a dollar ? 12. If 5 pounds of coffee cost if of a dollar, what will 1 pound cost ? 13. At $6 a barrel, what part of a barrel of flour can be bought for -f of a dollar ? 14. If 10 bushels of barley cost 3J dollars, what will 1 bushel cost ? NOTE. We reduce the mixed number to an improper fraction and divide as in the case of a simple fraction. OPERATION. J/-f-10 = i Ans. 15. If 21 pounds of raisms cost 4| dollars, what will 1 pound cost ? 16. If 12 men consume 6f pounds of meat in a day ; how much does 1 man consume ? 170 DIVISION OF 186. To divide a whole number by a fraction. I. At f of a dollar apiece, how many hats can be bought for 6 dollars ? ANALYSIS. Since of a dollar will OPERATION. buy one hat, 6 dollars will buy as many 6-=-4-= 6x5-f-4 = 7i. hats as is contained times in 6 ; and as there are 5 times as many fifths as whole things in any number, in 6 there are 30 fifths, and 4 fifths is contained in 2 ; 30 fifths 7i times : hence, _ Invert the terms of the divisor and multiply the whole num- ber by the new fraction. EXAMPLES. 1. Divide 14 by J. 3. Divide 63 by f . 2. Divide 212 by 4. Divide 420 by 5. At -^ of a dollar a yard, how many yards of cloth can be bought for 9 dollars ? 6. If a man travel ^ of a mile in 1 hour, how long will it take him to travel 10 miles ? 7. If y of a ton of hay is worth 9 dollars, what is a ton worth ? 187. To divide one fraction by another. 1. At f of a dollar a gallon, how much molasses can be bought for | of a dollar ? ANALYSIS. Since of a dollar OPERATION. will buy 1 gallon, I of a dollar will J-T-?-= I x 4^^4 -- - --- " ^ " buy as many gallons as \ is contained g times in \ : one is contained in I, I o times : but & is contained 5 times as many times as 1, or *- times ; but 2 161 fifths is contained half as many times as i, or f $ times, equal to 2- 1 3 -j times : hence, I. Invert the terms of the divisor. II. Multiply the numerators together for the numerator of the quotient, and the denominators together for the de- nominator of the quotient. 186. How do you divide a whole number by a fraction ? COMMON FRACTIONS. 171 NOTES. 1. If the vertical line is used, the denominator of the dividend and the numerator of the divisor fall 011 the left, and the other terms on the right. 2. Cancel all common factors. 3. If the dividend and divisor have a common denominator, they will cancel, and the quotient of their numerators will be the answer. 4. When the dividend or divisor contains a whole or mixed number, or compound fractions, reduce them to tiie form of simple fractions before dividing. EXAMPLES. 1. Divide -ft by ft. 2. Divide -ft by T f . 3. Divide 3 by |f 4. Divide } of f by T ^ of 1J. 5. Divide f of 21 by f of 3|. 6. Divide 6| by 2J. 7. At l of a dollar a pound, how much butter can be bought for | of a dollar ? 8. If 1 man consume 1^ pounds of meat in a day, how many men would 8J- pounds supply ? 9. If 6 pounds of tea cost 4J dollars, what does it cost a pound ? 10. At it of a dollar a basket, how many baskets of peaches can be bought for 11^ dollars ? 11. If of a ton of coal cost 6| dollars, what will 1 ton cost, at the same rate ? 12. How much cheese can be bought for -J of a dollar at of a dollar a pound ? 13. A man divided 2f dollars among his children, giving them y 7 ^ of a dollar a piece ; how many children had he ? 14. How many times will J-J- of a gallon of beer fill a vessel holding i of f gallons ? 15. How many tunes is of ^ of 27 contained in - of i of42? 16. If 5-J- bushels of potatoes cost 2f dollars, how much do they cost a bushel ? 17. If John can walk 21 miles in -^ of a day, how far can he walk in 1 day ? 18. If a turkey cost If dollars, how many can be bought for 12f dollars ? 19. At f of | of a dollar a yard, how many yards of rib- bon can be bought for -|i of a dollar ? 187. How do you divide one fraction by another 9 172 REDUCTION OF REDUCTION OP COMPLEX FRACTIONS. 188. Complex Fractions are only other forms of expression for the division of fractions : thus ; 1 is the same as % divided by -?j ; and may be written, % x f =f =2^-. 181). To reduce a complex fraction to the form of a sim- ple fraction. 1. Reduce _ to its simplest form. *i OPERATION. 4 ?j^--!=^xA =T s^ Ans.-, hence, 4 2 TJ- 09 3 RULE. Divide the numerator of the complex fraction by its denominator, Or : Multiply the numerator of the upper fraction into the denominator of the loiuer,for a numerator ; and the denomi- nator of the upper fraction into* the numerator of the lower, for a denominator. 9 NOTES. 1. When either of the terms of a complex fraction is a mixed number, or compound fraction, it must first be reduced to the form of a simple fraction. 2. When the vertical line is used, the numerator of the upper and the denominator of the lower numbers fall on the right of the verti- cal line, and the other terms on the left. EXAMPLES. Reduce the following complex fractions to their simplest form : 1. Reduce jL 2. Reduce ^1 3. Reduce 4. Reduce f of i. 5 Reduce 6. Reduce f. 8f 1. Reduce 8. Reduce __ * of 15 214f 25H 9. Reduce '^, 10 . Reduce of 48 DENOMINATE FRACTIONS. 173 DENOMINATE FRACTIONS. 190. A DENOMINATE Fraction is one in which the unit of the fraction is a denominate number. Thus, f of a yard is a denominate fraction. 191. REDUCTION of denominate fractions is the operation of changing a fraction from one denominate unit to another without altering its value. There are four cases : 1st. To change from a greater unit to a less, as from yards to inches : 2d. To change from a less unit to a greater : 3d. To find the value of a fraction in integers of lower denominations : 4th. To find the value of integers in a fraction of a larger unit. These cases will be arranged in sets of two and two. 192. To change from a greater unit to & less. 1. In $ of a yard, how many inches ? OPERATION. f x 3 x 12=if=20 inches. ANALYSIS. Since in 1 yard there are 3 feet, in f yards there are $ times 3 feet=-^- feet. And since in 1 foot there are 12 inches, in ^ feet there are 1 9 - times 12 inches = I a = 20 inch's : hence, RULE. Multiply the frac- tion and the products which arise by the units of the scale, in succession, until you reach the unit required. 193. To change from a less unit to a greater. 1. In 20 inches, how many yards ? OPERATION. 20 xAxi=H=* J ards < ANALYSIS. Since 12 inches make 1 foot, in 20 inches there are as many feet as 12 inches is contained times in 20 inches = H feet; and as 3 feet make 1 yard, in ^ feet there are as many yards as 3 feet is contained times in ^ fect=|=f yards: hence, RULE. Divide the fraction and the quotients which arise, by the units of the scale, in suc- cession, until you reach the unit required. 188. What are complex fractions? 189. How do you reduce complex to simple fractions ? 174 DENOMINATE FRACTIONS. NOTE. In every operation of reduction, in which there are common factors, be sure and cancel them before making the final multiplication. EXAMPLES. 1. Reduce -g-f-g- of a hogshead to the fraction of a quart. 2. Reduce -^ of a bushel to the fraction of a pint. 3. Reduce -g^ir of a pound Troy to the fraction of a grain. 4. What part of a foot is -J-&TF of a furlong ? 5. What part of a minute is -^Vo- f a day ? 6. Reduce ^Vjizr f a cwt. to the fraction of an ounce. 7. Reduce f of a gallon to the fraction of a hogshead. 8. What part of a is of a shilling ? 9. What part of a hogshead is -g- of a quart ? 10. What part of a mile is -fr of a foot ? 11. Reduce 4-^0 of to the fraction of a farthing. 12. Reduce yV of an Ell Eng. to the fraction of a nail. 13. Reduce |- of a nail to the fraction of a yard ? 14. Reduce J of % of a foot to the fraction of a mile. 15. Reduce 5 ^ 7 6 of a ton to the fraction of a pound. 16. Reduce J[ of 3| pwt. to the fraction of a pound Troy. It. What part of a mile is j of a rod ? 18. What part of an ounce is -fo of a scruple ? 19. -^f-g- of a day is what portion of 10 minutes? 20. What part of J- of a foot is yf-g- of a furlong ? 21. Reduce -g^g- of a hogshead of ale to the fraction of a pint. 190. What is a denominate fraction ? 191. What is reduction of denominate fractions? How many casca are there V Name them. 192. How do you change from a greater unit to a less ? 193. How do you change from a less unit to a greater ? DENOMINATE FRACTIONS. 175 194. To find the value of a fraction in integers of loiver denominations. 1. What is the value of f of a pound Troy ? ANALYSIS. of a pound re- duced to the fraction of an ounce is |xl2=^. of an ounce, (Art. 177.), which is equal to 9- ounces : f of an ounce reduced to the fraction of a pennyweight is | x 20=^ of a pwt., or 12pwt. OPERATION. burner. 4 12 oz. pwt. Denom. 5)48(9... 12 45 3 20 5)60 60 RULE. I. Multiply the numerator of the fraction by the number which will re- duce it to the next lower de- nomination and divide the product by the denominator. II. If there is a remain- der, reduce it in the same manner, and so on, till the lowest denomination is obtained. 195. To find ike value of integers in a fraction of a higher denomination. 2. Reduce 9oz. 12pwts. to the fraction of a pound Troy. ANALYSIS. In 1 pound there are 240 pennyweights: 1 pen- ny weight is ^ of a pound ; and 9 ounces 12pwts. = l&Zpwts. is of a pound= of a pound. OPERATION. 1 lb. oz. pwts. 12 9.. 12 12 20 20 Num - l92_ 240 Denom. 40" ~ RULE. I. Reduce the given ntegers to the lowest de- nomination named, and the result will be the numerator jf the required fraction. II. Eeduce 1 unit of the required denomination, to the denomination of the numera- or, and the result will be he denominator of the re- quired fraction. EXAMPLES. 3. What is the value of - of a tun of wine ? 4. What part of a tun of wine is 3hhd. Slgal. 2gt. ? 194. How do you find the value of a fraction in integers of lower de- nominations ? 195. How do yon find the value of integers in a fraction of a higher denomination ? 176 ADDITION AND SUBTRACTION OF 5. What is the value of y 9 ^ of a yard ? 6. What is the value of -| of a month ? 7. What is the value of f of a chaldron ? 8. What is the value of % of a mile ? 9. What is the value of -fe of a ton ? 10. What is the value of $ of 3 days ? 11. What is the value of of of 6 bushels of grain ? 12. Reduce Sgals. 2qts. to the fraction of a hogshead. 13. Reduce 2fur. 36rd 2yd. to the fraction of a mile. 14. What part of a is 5s. *Id. ? 15. What part of a pound Troy is lOoz. 13pwt. Sgr. ? 16. llcwt. Qqr. 12/6. 7 02. l%dr. is what part of a ton? 17. What part is 2pk. qt. of Ibu. Spk. ? 18. 24/6. 6oz. is what part of Zqr. 12/6. I2oz. ? 19. Reduce 3wk. Id. 9/i. 36?n. to the fraction of a month 20. Reduce 2E. 32rrf. 8z/<7. to the fraction of an acre. 21. Reduce 12s. $d. \\far. to the fraction of a guinea. 22. What is the value of T y&, apothecaries' weight ? 23. What part of an Ell English is 3qr. 3?ia. l\in. ? 24. What is the value of $hhd. ? 25. What is the value gf f of 3 barrels of beer ? 26. What is the value of T V of a cwt. ? 27. Reduce 3 15' 18|" to the fraction of a sign. 28. Reduce 3 inches to the fraction of a hand. 29. What is the value of -fa of a hogshead of wine ? 30. What is the value of 7 of an acre of land ? ADDITION AND SUBTRACTION. 196. To add or subtract denominate fractions. 1. Add of a to of a shilling. | of a = of 2^=*$- of a shilling. Then, 4J* + f ^W+lf =W*= * = 14s 2 ^ 196. Give the rule for adding and subtracting denominate fractions. DENOMINATE FRACTIONS. 177 Or, the |- of a shilling may be reduced to the fraction of a > : thus, I f ^V=Tth> of a &=& of a : then, S+A = H+A=H of a > which being reduced, gives 14s. %d. Ans. 2. Add f of a year, | of a week, and | of a day. f of a year=f of -^p days=31w&. 2da. J of a week=J of 7 days - - 2da. Shr. I of a day = - - - - = - - - 3/tr. Ans. Slwk. Ida. llhr. 3. From \ of a take J of a shilling. J of a shilling^ of -5^ of a =-fa of a . Then, ' i AF=^-A=-ofa^=9- 8 ^ 4. From 1 j#>. Troy weigfit, take ^oz. Ib. oz. pwt. gr. lJ/6.= of Jjao2=21oz. = l 9 Joz.=^ of-y- ofygrr. = 80gfr. = 038 J?is. 1 8 16 16 RULE. Reduce the given fractions to the same unit, and then add or subtract as in simple fractions, after ivhich reduce to integers of a lower denomination : Or : Reduce the fractions separately to integers of lower de- nominations, and then add or subtract as in denominate num" bers. EXAMPLES. 5. Add 1J miles, T ^ furlongs, and 30 rods. 6. Add of a yard, J of a foot, and $ of a mile. 7. Add | of a cwt., * of a Ib., 13oz., J of a curt, and 6/6. 8. From J of a day take f of a second. 9. From | of a rod take f of an inch. 10. From *fc of a hogshead take f of a quart. 11. From $oz. take %pwl. 12. From 4fcw. take 4 T y&. 12 178 DUODECIMALS. 13. Mr. Merchant bought of farmer Jones 22J bushels of wheat at one time, 19^ bushels at another, and 33f at an- other : how much did he buy in all ? 14. Add % of a ton and -fa of a cwt. 15. Mr. Warren pursued a bear for three successive days ; the first day he travelled 28-f- miles ; the second 33 T ^ miles ; the third 29-^j- miles, when he overtook him : how far had he travelled ? 16. Add 5f days and 52 T %- minutes. 17. Add $cwt., S%lb., and 3 T y&. 18. A tailor bought 3 pieces of cloth, containing respect- ively, 18| yards, 21| Ells Flemish, and 16f Ells English : how many yards in all ? 19. Bought 3 kinds of cloth ; the first contained \ of 3 of f of yards ; the second, of f of 5 yards ; and the third, \ of f of | yards : how much in them all ? 20. Add \\cwt. 17f/&. and 7foz. 21. From f of an oz. take of &pwt. 22. Take } of a day and J of of j of an hour from 3 1 weeks. 23. A man is 6| miles from home, and travels 4wi. Ifur. 24?*d., when he is overtaken by a storm : how far is he then from home ? 24. A man sold -J^ of his farm at one time, ^ at another, and ^7 at another : what part had he left ? 25. From 1 J of a take | of a shilling. 26. From loz. take %pwt. 27. From 8%cwt. take 4 T y6. 28. From 3|Z6. Troy weight, take \pz. 29. From 1^ rods take ^ of an inch. 30. From $f g) take ^ 3 . DUODECIMALS. 197. If the unit 1 foot be divided into 12 equal parts, each part is called an inch or prime, and marked '. If an inch be divided into 12 equal parts, each part is called a second, and marked ". If a second be divided, in like manner, into 12 DUODECIMALS. 179 equal parts, each part is called a third, and marked "' ; and so on for divisions still smaller. This division of the foot gives 1' inch or prime - - , - - - = -^ of a foot. I" second is ^ of & - - - = y^ of a foot. 1'" third is T V of & of A' - = TT^ of a foot - NOTE. The marks ', ", '", &c., which denote the fractional units, are called indices, TABLE. 12'" make 1" second. 12" " 1' inch or prime. 12' " 1 foot. Hence : Duodecimals are denominate fractions, in which the primary unit is 1 foot, and 12 the scale of division. NOTE. Duodecimals are chiefly used in measuring surfaces and solids. ADDITION AND SUBTRACTION. 198. The units of duodecimals are reduced, added, and subtracted, like those of other denominate numbers. The scale is always 12. EXAMPLES. 1. In 185', how many feet ? 2. In 250", how many feet and inches ? 3. In 4367'", how many feet? 4. What is the sum of 3/35. 6' 3" 2'" and 2ft. I' 10" 11'"? 5. What is the sum of 8/3L 9' 7" and 6/fc. 7' 3" 4"' ? 6. What is the difference between 9/fc. 3' 5" 6'" and 7/35. 3' 6" 7'"? 7. What is the difference between 40/35. 6' 6" and 29/fc. 7'" ? 8. What is the difference between 12ft. 7' 9" 6'" and 4/2. 9' 7" 9'"? 197. If 1 foot be divided into twelve equal parts, what is each part called ? If the inch be so divided, what is each part called ? What are duodecimals ? For what are duodecimals chiefly used ? 198. How do you add and subtract duodecimals ? What is the scale ? 180 DUODECIMALS. MULTIPLICATION. 199. Begin with the highest unit of the multiplier and the lowest of the multiplicand, and recollect, 1st. That 1 foot x 1 foot=l square foot (Art. 110). 2d. That a part of a foot x a part of a foot = some part of a square foot. NOTE. Observe that the unit is changed, by multiplication, from a linear to a superficial unit. Multiply 6ft. T 8" by 2/fc. 9'. OPERATION. ANALYSIS. Since a prime is ^ of a ft. foot and a second T^T, g y g" 2 x 8" =-i i A of a square foot ; which re- 9 Q / duced to 12ths, is 1' and 4" : that is, 1 twelfth, and 4 twelfths of -fe of a 2 X 8"= 1' 4" square foot. 2x7'= 1 2' 2x7' =14 twelfths=l/. 2' 2 X 6 =12 2x6 =12 square feet, 9' x g" 6" 9 x 8"= T ^|-8 of a square foot=6" 9' x 7' 5' 3" 9'xT=fA-=5' 3" 9' X 6 = 4 6' 9 x6'=f|=46' p rod 18 3' r RULE. I. Write the multiplier under the multiplicand, so that units of the same order shall fall in the same column. II. Begin with the highest unit of the multiplier and the lowest of the multiplicand, and make the index of each product equal to the sum of the indices of the factors. III. Eeduce each product, in succession, to the next higher denomination, when possible. NOTE. The index of the unit of any product is equal to the of the indices of the factors. EXAMPLES. 1 . How many solid feet in a stick of timber which is 25 feet 6 inches long, 2 feet 7 inches broad, and 3 feet 3 inches thick ? 199. Explain the method of multiplying duodecimals. Give the rule. DUODECIMALS. 181 OPERATION. .# Beginning with the 2 feet, we say 2 25 6' length, times 6' are 12'=1 square foot : then, 2 27' breadth. times 25 are 50, and 1 to carry are 51 f square feet. 51 Next, 7 times 6' are 42", =3' and 6" : 3' 6' then 7' times 25=175'=14 7': hence, the ^4 j' surface is 65 10' 6", and by multiplying by the thickness, we find the solid contents 65 1" o to be 214 1' 1" 6'" cubic feet. 3 X thickness. 197 7' 6" 16 5' 7" 6"'' 214 1'1"6'" 2. Multiply 9/2. 4m. by 8/2. 3m. 3. Multiply 9#. 2m. by fyfc 6m. 4. Multiply 24/2. 10m. by 6/2. 8m. 5. Multiply 70/2. 9m. by 12/2. 3m. 6. How many cords and cord feet in a pile of wood 24 feet long, 4 feet wide, and 3 feet 6 inches high ? 7. How many square feet are there in a board 17 feet 6 inches in length, and 1 foot 7 inches in width ? 8. What number of cubic feet are there in a granite pillar 3 feet 9 inches in width, 2 feet 3 inches in thickness, and 12 feet 6 inches in length ? 9. There is a certain pile of wood, measuring 24 feet in length, 16 feet 9 inches high, and 12 feet 6 inches in width. How many cords are there in the pile ? 10. How many square yards in the walls of a room, 14 feet 8 inches long, 11 feet 6 inches wide, and 7 feet 11 inches high ? 11. If a load of wood be 8 feet long, 3 feet 9 inches wide, and 6 feet 6 inches high, how much does it contain ? 12. How many cubic yards of earth were dug from a cellar which measured 42 feet 10 inches long, 12 feet 6 inches wide, and 8 feet deep ? 13. What will it cost to plaster a room 20 feet 6' long, 15 feet wide, 9 feet 6' high, at 18 cents per square yard? 14. How many feet of boards 1 inch thick can be cut from a plank 18/2. 9m. long, l/t. Sin. wide, and 3m. thick, if there is no waste in sawing ? 182 DECIMAL FRACTIONS. DECIMAL FRACTIONS. 200. There are two kinds of Fractions : Common Frar tions and Decimal Fractions. A Common Fraction is one in which the unit is divided into any number of equal parts. A Decimal fraction is one in which the unit is divided ac- cording to the scale of tens. 201. If the unit 1 be divided into 10 equal parts, the parts are called tenths. If the unit 1 be divided into one hundred equal parts, the parts are called hundredths. If the unit 1 be divided into one thousand equal parts, the parts are called thousandths, and we have similar expressions for the parts, when the unit is further divided according to the scale of tens. * These fractions may be written thus : Four-tenths, ----- *fo. Six-tenths, - - T V Forty-five hundredths, 125 thousandths, 1047 ten thousandths, - From which we see, that in each case the denominator indicates the fractional unit ; that is, determines whether it is one-tenth, one-hundredth, one-thousandth, &c. 202. The denominators of decimal fractions are seldom written. The fractions are usually expressed by means of a period, placed at the left of the numerator. Thus ^5- is written - . 4 200. How many kinds of fractions are there? What are they? What is a common fraction ? What is a decimal fraction ? 201. When the unit 1 Is divided into 10 equal parts, what is each part called ? What is each part called when it is divided into 100 equal parts? When into 10000? Into 10,000, &c. ? How are decimal frac- tions formed ? What gives denomination to the fraction ! DECIMAL FRACTIONS. 183 This method of writing decimal fractions is" a mere lan- guage, and is used to avoid writing the denominators. The denominator, however, of every decimal fraction is always understood : It is the unit 1 with as many ciphers annexed as there are places of figures in the decimal. The place next to the decimal point, is called the place of tenths, and its unit is 1 tenth. The next place, to the right, is the place of hundredths, and its unit is 1 hundreth ; the next is the place of thousandths, and its unit is 1 thous- andth ; and similarly for places still to the right. DECIMAL NUMERATION TABLE. d S T3 2 | oJ 'o a 2 'O'fsS 'g , Sg^ rS "3 a -*' ^ 2 a a .4 is read 4 tenths, .54 - - 54 hundredths. .064 - - 64 thousandths. .6754 - - 6154 ten thousandths, .01234 - - 1234 hundred thousandths .007654 - - 7654 mfflionths. .0043604 - - 43604 ten millionths. NOTE. Decimal fractions are numerated from left to right ; thus, tenths, hundredths, thousandths, &c. 202. Are the denominators of decimal fractions generally written ? How are the fractions expressed? Is the denominator understood?. What is it ? What is the place next the decimal point called ? What is its unit ? What is the next place called ? What is its unit ? What is the third place called ? What is its unit ? Which way are decimals numerated ? 184 DECIMAL FRACTIONS. 203. Wfite and numerate the following decimals : Four tenths, .4 Four hundredths, - .0 4 Four thousandths, .004 Four ten thousandths, - .0004 Four hundred thousandths, .00004 Four millionths, - .000004 Four ten millionths, .0000004. Here we see, that the same figure expresses different deci- mal units, according to the place which it occupies : therefore, The value of the unit, in the different places, in passing from the left to the right, diminishes according to the scale of tens. Hence, ten of the units in any place, are equal to one unit in the place next to the left ; that is, ten thousandths make one hundredth, ten hundredths make one-tenth, and ten-tenths, the unit 1. This scale of increase, from the right hand towards the left, is the same as that in whole numbers ; therefore, Whole numbers and decimal fractions may be united by placing the decimal point between them : thus, Whole numbers. Decimals. I I 836 3'0 641. 0478976 A number composed partly of a whole number and partly of a decimal, is called a mixed number. DECIMAL FRACTIONS. 185 RULE FOR WRITING DECIMALS. Write the decimal as if it were a whole number, prefix- ing as many ciphers as are necessary to make it of the required denomination. RULE FOR READING DECIMALS. Read the decimal as though it were a whole number, adding the denomination indicated by the lowest decimal unit. EXAMPLES. Write the following numbers, decimally : (1.) (2.) (3.) (4.) (5.) 3 16 17 32 165 10 , 1000 10000 100 10000 (6.) (7.) (8.) (9.) (10.) Write the following numbers in figures, and then numerate them. 1. Forty-one, and three-tenths. 2. Sixteen, and three millionths. 3. Five, and nine hundredths. 4. Sixty-five, and fifteen thousandths. 5. Eighty, and three millionths. 6. Two, and three hundred millionths. 7. Four hundred, and ninety-two thousandths. 8. Three thousand, and twenty-one ten thousandths. 9. Forty-seven, and twenty-one hundred thousandths. 10. Fifteen hundred, and three millionths. 11. Thirty-nine, and six hundred and forty thousandths. 12. Three thousand, eight hundred and forty millionths. 1 3. Six hundred and fifty thousandths. 203. Docs the value of the unit of a figure depend upon the place which it occupies V How does the value change from the left towards the right ? What do ten units of any one place make ? How do the units of the place increase from the right towards the left ? How may whole numbers be joined with decimals? What is such a number called? Give the rule for writing decimal fractions. Give the rule for reading decimal fractions. 186 UNITED STATES MONEY. UNITED STATES MONEY. 204. The denominations of United States Money correspond to the decimal division, if we regard 1 dollar as the unit. For, the dimes are tenths of the dollar, the cents are hun- dredths of the dollar, and the mills, being tenths of the cent, are thousandths of the dollar. EXAMPLES. 1. Express $39 and 39 cents and 7 mills, decimally. 2. Express $12 and 3 mills, decimally. 3. Express $147 and 4 cents, decimally. 4. Express $148 4 mills, decimally. 5. Express $4 6 mills, decimally. 6. Express $9 6 cents 9 mills, decimally. 7. Express $10 13 cents 2 mills, decimally. ANNEXING AND PREFIXING CIPHERS. 205. Annexing a cipher is placing it on the right of a number. If a cipher is annexed to a decimal it makes one more deci- mal place, and therefore, a cipher must also be annexed to the denominator (Art. 202). The numerator and denominator will therefore have been multiplied by the same number, and consequently the value of the fraction will not be changed (Art. 161) : hence, Annexing ciphers to a decimal fraction does not alter its value. We may take as an example, .3 T 3 7 . If we annex a cipher, to the numerator, we must, at the same time, annex one to the denominator, which gives, 204. If the denominations of Federal Money be expressed decimally what is the unit ? What part of a dollar is 1 dime ? What part of a dime is a eent ? What part of a cent is a mill ? What part of a dollar is 1 cent ? 1 mill ? 305. When is a cipher annexed to a number? Does the annexing of ciphers to a decimal alter its value ? Why not ? What dp three tenths become by annexing a cipher ? What by annexing two ciphers ? Three ciphers? What do 8 tenths become by annexing a cipher? By annexing two ciphers V By annexing three ciphers t DECIMAL FRACTIONS. 187 ,3 = -j^j- = .30 by annexing one cipher, .3 = T 3 TM7ir -300 by annexing two ciphers. if a decimal point be placed on the right of an integral number, and ciphers be then annexed, the value will not be changed : thus, 5 = 5.0 = 5.00 = 5.000, &c. 206. Prefixing a cipher is placing it on the left of a number. If ciphers are prefixed to the numerator of a decimal frac- tion, the same number of ciphers must be annexed to the denominator. Now, the numerator will remain unchanged while the denominator will be increased ten times for every cipher annexed ; and hence, the value of the fraction will be diminished ten times for every cipher prefixed to the nume- rator (Art. 160). Prefixing ciphers to a decimal fraction diminishes its value ten times for every cipher prefixed. Take, for example, the fraction .2= T *j-. .2 becomes -ffc = .02 by prefixing one cipher, .2 becomes -fipfc = - 002 by prefixing two ciphers, .2 becomes -ffiPfc = .0002 by prefixing three ciphers : in which the fraction is diminished ten times for every cipher prefixed. ADDITION OF DECIMALS. 207. It must be remembered, that only units of the same kind can be added together. Therefore, in setting down decimal numbers for addition, figures expressing the same unit must be placed in the same column. 200. When is a cipher prefixed to a number ? When prefixed to a decimal, does it increase the numerator ? Does it increase the denomi- nator? What effect then has it on the value of the fraction ? What do .3 become by prefixing; a cipher? By prefixing two ciphers? By prefixing three? What do .07 become by prefixing a cipher ? By pre- fixing two ? By prefixing three ? By prefixing four ? 207. What parts of unity may be added together ? How do you set down the numbers for addition? How will the decimal points fall ? How do you then add ? How many decimal places do you point off m the sum ? 188 ADDITION OF The addition of decimals is then made in the same manner is that of whole numbers. I. Find the sum of 37.04, 704.3, and .0376. OPERATION. Place the decimal points in the same column : HA this brings units of the same value in the same 704.3 column : then add as in whole numbers : hence, .0376 741.3776 RULE. I. Set down the numbers to be added so that figures of the same unit value shall stand in the same column. II. Add as in simple numbers, and point off in the sum from the right hand, as many places for decimals as are equal to the greatest number of places in any of the numbers added. PROOF. The same as in simple numbers. EXAMPLES. 1. Add 4.035, 763.196, 445.3741, and 91.3754 together. 2. Add 365.103113, .76012, 1.34976, .3549, and 61.11 together. 3. 67.407 + 97.004+4 + .6 + .06 + .3. 4. .0007 + 1.0436 + .4 + .05 + .047. 5. .0049+47.0426 + 37.0410 + 360.0039. 6. What is the sum of 27, 14, 49, 126, 999, .469, and .2614 ? 7. Add 15, 100, 67, 1, 5, 33, .467, and 24.6 together, 8. What is the sum of 99, 99, 31, .25, 60.102, .29, and 100.347? 9. Add together .7509, .0074, 69.8408, and .6109. 10. Required the sum of twenty-nine and 3 tenths, four hundred and sixty-five, and two hundred and twenty-one thousandths. 1 1 . Required the sum of two hundred dollars one dime three cents and 9 mills, four hundred and forty dollars nine mills, and one dollar one dime and one mill. 12. What is the sum of one-tenth, one hundredth, and one thousandth ? DECIMAL FRACTIONS. 189 13. What is the sum of 4, and 6 ten-thousandths ? 14. Required, in dollars and decimals, the sum of one dollar one dime one cent one mill, six dollars three mills, four dol- lars eight cents, nine dollars six mills, one hundred dollars six dimes, nine dimes one mill, and eight dollars six cents. 15. What is the sum of 4 dollars 6 cents, 9 dollars 3 mills, 14 dollars 3 dimes 9 cents 1 mill, 104 dollars 9 dimes 9 cents 9 mills, 999 dollars 9 dimes 1 mill, 4 mills, 6 mills, and 1 mill? 16. If you sell one piece of cloth for $4,25, another for $5,075, and another for $7,0025, how much do you get for all? 17. What is the amount of $151,7, $70,602, $4,06, and $807,2659 ? 18. A man received at one time $13,25 ; at another $8,4 ; at anotlier $23,051j at another $6 ; and at another $0,75 : how much did he receive in all ? 19. Find the sum of twenty-five hundredths, three hundred and sixty-five thousandths, six tenths, and nine millionths. 20. What is the sum of twenty-three millions and ten, one thousand, four hundred thousandths, twenty-seven, nineteen millionths, seven and five tenths ? 21. What is the sum of six millionths, four ten-thousandths, 19 hundred thousandths, sixteen hundredths, and four tenths? 22. If a piece of cloth cost four dollars and six mills, eight pounds of coffee twenty-six cents, and a piece of muslin three dollars seven dimes and twelve mills, what will be the cost of them all ? 23. If a yoke of oxen cost one hundred dollars nine dimes and nine mills, a pair of horses two hundred and fifty dollars five dimes and fifteen mills, and a sleigh sixty-five dollars eleven dimes and thirty-nine mills, what will be their entire cost? 24. Find the sum of the following numbers : Sixty-nine thousand and sixty-nine thousandths, forty-seven hundred and forty-seven thousandths, eighty-five and eighty-five hun- dredths, six hundred and forty-nine and six hundred and forty-nine ten-thousandths ? 100 SUBTRACTION OF SUBTRACTION OF DECIMALS 208. Subtraction of Decimal Fractions is the operation of finding the difference between two decimal numbers. I. From 3.275 to take .0879. NOTE. In this example a cipher is annexed OPSBATION. to the minuend to make the number of decimal 3.2750 places equal to the number in the subtrahend. This 08 *7 Q does not alter the value of the minuend (Art. 205) hence, 3.1871 RULE. I. Write the less number under the greater, so that figures of the same unit value shall stand in the same column. II. Subtract as in simple numbers, and point off the deci- mal places in the remainder, as in addition. PROOF. Same as in simple numbers. EXAMPLES. 1. From 3295 take .0879. 2. From 291.10001 take 41.375. 3. From 10.000001 take 111111. 4. From 396 take 8 ten-thousandths. 5. From 1 take one thousandth. 6. Fcom 6378 take one-tenth. 7. From 365.0075 take 3 millionths. 8. From 21.004 take 97 ten-thousandths. 9. From 260.4709 take 47 ten-millionths. 10. From 10.0302 take 19 millionths. 11. From 2.01 take 6 ten-thousandths. 12. From thirty-five thousands take thirty-fire thousandths. 13. From 4262.0246 take 23.41653. 14. From 346.523120 take 219.691245943. ' 15. From 64.075 take .195326. 16. What is the difference between 107 and .0007? 17. What is the difference between 1.5 and .3785 ? 18. From 96. 71 take 96.709. 208. What is subtraction of decimal fractions ? How do you set down the numbers for subtraction ? How do you then subtract ? How many decimal places do you point off in the remainder ? DECIMAL FRACTIONS. 191 MULTIPLICATION OF DECIMAL FRACTIONS. 209. To multiply one decimal by another. 1. Multiply 3.05 by 4.102. OPERATION. ANALYSIS. If we change both factors to vul- s. 3 05 &r fractions, the product of the numerator will 4JJL2. 1 1Q9 be the same as that of the decimal numbers, and the number of decimal places will be equal to the 610 number of ciphers in the two denominators: 305 hence, 12 . 20 12.51110 RULE. Multiply as in simple numbers, and point off" in the product, from the right hand, as many figures for decimals as there'are decimal places in both factors ; and if there be not so many in the product, supply the deficiency by prefixing ciphers. EXAMPLES 1. Multiply 3. 049 by .012. 2. Multiply 365.491 by .001. 3. Multiply 496. 0135 by 1.496. 4. Multiply one and one milliouth by one thousandth. 5. Multiply one hundred and forty-seven millionths by one millionth. 6. Multiply three hundred, and twenty-seven hundredth^ by 31. 7. Multiply 31.00467 by 10.03962. 8. What is the product of five-tenths by five-tenths ? 9. What is the product of five-tenths by five-thousandths ? 10. Multiply 596.04 by 0.00004. 11. Multiply 38049.079 by 0.00008. 12. What will 6.29 weeks' board come to at 2.75 dollars per week ? 13. What will 61 pounds of sugar come to at $0.234 per pound ? 209. After multiplying, how many decimal places will you point off In the product ? When there are not so many in the product what do you do ? Give the rule for the multiplication of decimals. 192 CONTRACTIONS. 14. If 12 . 836 dollars are paid for one barrel of flour, what will . 354 barrels cost ? 15. What are the contents of a board, . 06 feet long and . 06 wide? 16. Multiply 49000 by .0049. 17. Bought 1234 oranges for 4 . 6 cents apiece : how much did they cost ? 18. What will 375.6 pounds of coffee cost at .125 dollars per pound ? 19. If I buy 36. 251 pounds of indigo at $0.029 per pound, what will it come to ? 20. Multiply $89. 3421001 by .0000028. 21. Multiply $341.45 by .007. 22. What are the contents of a lot which is . 004 miles long and . 004 miles wide ? 23. Multiply .007853 by .035. 24. What is the product of $26.000375 multiplied 1>v .00007? CONTRACTIONS. 210. When a decimal number is to be multiplied by 10, 100, 1000, &c., the multiplication may be made by removing the decimal point as many places to the right hand as there are ciphers in the multiplier, and if there be not so many figures on the right of the decimal point, supply the deficiency by annexing ciphers. Thus, 6.79 multiplied by - 10 100 1000 10000 100000 Also, 370 . 036 multiplied by flO 1 |100 1000 L = 10000 100000 J 67.9 679 6790 67900 679000 3700.36 37003.6 370036 3700360 37003600 210. How do you multiply a decimal number by 10, 100, 1000, Ac. ? If there are not as many decimal figures as there are ciphers in the multiplier, what do you <lo ? DECIMAL FRACTIONS. 193 DIVISION OF DECIMAL FRACTIONS. 211. Division of Decimal Fractions is similar to that of simple numbers. 1. Let it be required to divide 1.38483 by 60.21. ANALYSIS. The dividend must be equal OPERATION. to the product of the divisor and quotient, 60 . 21 ) 1 . 38483(23 (Art, 61) ; and hence must contain as j 2042 many decimal places as both of them ; therefore, There, must be as many decimal places in 18063 the quotient as the decimal places in the divi- ~r~ 7\wi dend exceed those in the divisor : hence, R.ULE. Divide as in simple numbers, and point off" in the quotient, from the right hand, as many places for decimals as the decimal places in the dividend exceed those in the divisor ; and if there are not so many, supply the deficiency by prefix- ing ciphers. EXAMPLES. 1. Divide 2.3421 by 2.11 2. Divide 12.82561 by 3.01. 3. Divide 33.66431 by 1.01. 4. Divide .010001 by .01. 5. Divide 8.2470 by .002. 6. Divide 94.0056 by .08. 7. What is the quotient of 37 . 57602, divided by 3 ; by . 3 ; by .03; by .003; by .0003? 8. What is the quotient of 129.75896, divided by 8 ; by .08; by .008; by .0008; by .00008? 9. What is the quotient of 187 .29900, divided by 9 ; by .9 ; by .09 ; by .009 ; by .0009 ; by .00009 ? 10. What is the quotient of 764 2043244, divided by 6 ; by .06 ; by .006 ; by .0006 ; by .00006 ; by .000006? NOTE. 1. When there are more decimal places in the divisor than in the dividend, annex ciphers to the dividend and make the decimal places equal ; all the figures of the quotient will then be whole numbers. 211. How docs the number of decimal places in the dividend com- pare with that in the divisor and quotient? How do you determine the number of decimal places in the quotient? If the divisor contains four places and the dividend six, how many in the quotient ? If the divisor contains three places and the dividend five, how many in the quotient ? Give the rule for the division of decimals. 13 DIVISION OF EXAMPLES. 1. Divide 4397. 4 by 3. 49. NOTE. We annex one to the dividend. Had it contained no decimal place we should have annexed two. OPERATION. 3.49)4397.40(1260 349 907 698 2094 2094 An*. 1260. 2. Divide 2194.02194 by .100001. 3. Divide 9811. 0047 by .325947. 4. Divide .1 by .0001. | 5. Divide 10 by .15. 6. Divide 6 by .6 ; by .06 ; by. 006 ; by .2 ; by .3 ; by .003; by .5; by .05; by .005. NOTE. 2. When it is necessary to continue the division farther than the figures of the dividend will allow, we annex ciphers, and consider them as decimal places of the dividend. When the division does not terminate, we annex the plus sign to show that it may be continued : thus .2 divided by ^=.666+. EXAMPLES. 1. Divide 4. 25 by 1.25. ANALYSIS. In this example we annex one 0. and then the decimal places in the dividend will exceed those in the divisor by 1. OPERATION. 25)4.25(3.4 3.75 ~500 500 Ans. 3.4. 2. Divide . 2 by .6. 3. Divide 37. 4 by 4. 5. 4. Divide 586.4 by 375. 5. Divide 94 . 0369 by 81 . 032. NOTE. 3. When any decimal number is to be divided by 10, 100, 1000, &c., the division is made by removing the decimal point as many places to the left as there are Q's in Vie divisor ; and if there be not so many figures on the left of the decimal point, the deficiency is supplied by prefixing ciphers. 27 . 69 divided by 10 100 1000 10000 2.769 .2769 .02769 .002769 DECIMAL FRACTIONS. 195 10 100 642.89 divided by -I 1000 10000 100000 64.289 6.4289 .64289 .064289 .0064289 QUESTIONS IN THE PRECEDING RULES 1. If I divide .6 dollars among 94 men, how much will each receive ? 2. I gave 28 dollars to 267 persons : how much apiece ? 3. Divide 6 35 by .425. 4. What is the quotient of $36.2678 divided by 2.25 ? 5. Divide a dollar into 12 equal parts. 6. Divide .25 of 3.26 into .034 of 3.04 equal parts. 7. How many times will .35 of 35 be contained in .024 of 24? * 8. At .75 dollars a bushel, how many bushels of rye can be bought for 141 dollars ? 9. Bought 12 arid 15 thousandths bushels of potatoes for 33 hundredths dollars a bushel, and paid in oats at 22 hun- dredths of a dollar a bushel : how many bushels of oats did it take? 10. Bought 53.1 yards of cloth for 42 dollars : how much was it a yard ? 11. Divide 125 by .1045. 12. Divide one millionth by one billionth. 1 3. A merchant sold 4 parcels of cloth, the first contained 127 and 3 thousandths yards ; the 2d, 6 and 3 tenths yards ; the 3d, 4 and one hundredth yards ; the 4th, 90 and one millionth yards : how many yards did he sell in all ? 14. A merchant buys three chests of tea, the first contains 60 and one thousandth pounds ; the second, 39 and one ten thousandth pounds ; the third, 26 and one tenth pounds : how much did he buy in all ? NOTE. 1. If there are more decimal places in the divisor than in the dividend, what do you do ? What will the figures of the quotient then be? 2. How do you continue the division after you have brought down all the figures of the dividend ? What sign do you place after the quo- tient ? What does it show? 3. How do you divide a decimal fraction by 10, 100, 1000, &c. ? 19G DIVISION OF 15. What is the sum of $20 and three hundredths ; $4 and one-tenth, $6 and one thousandth, and $18 and one hundredth ? 16. A puts in trade $504.342 ; B puts in $350.1965 ; C puts in $100.11; D puts in $99.334; and E puts in $9001.32 : what is the whole amount put in ? It. B has $936, and A has $1, 3 dimes and 1 mill : how much more money has B than A ? 18. A merchant buys 37.5 yards of cloth, at one dollar twenty-five cents per yard : how much does the whole come to ? 19. If 12 men had each $339 one dime 9 cents and 3 mills, what would be the total amount of their money ? 20. A farmer sells to a merchant 13.12 cords of wood at $4.25 per cord, and 13 bushels of wheat at $1.06 per bushel : he is to take in payment 13 yards of broadcloth at $4.07 per yard, and the remainder in cash : how much money did he receive ? 21. If one man can remove 5.91 cubic yards of earth in a day, how much could nineteen men remove ? 22. What is the cost of 8.3 yards of cloth at $5.47 per yard? 23. If a man earns one dollar and one mill per day, how much will he earn in a year of 313 working days ? 24. What will be the cost of 375 thousandths of a cord of wood, at $2 per cord ? 25. A man leaves an estate of $1473.194 to be equally divided among 12 heirs : what is each one's portion ? 26. If flour is $9.25 a barrel, how many barrels can I buy for $1637.25 ? 27. Bought 26 yards of cloth at $4.37| a yard, and paid for it in flour at $7.25 a barrel : how much flour will pay for the cloth ? 28. How much molasses at 22|- cents a gallon "must be given for 46 bushels of oats at 45 cents a bushel? 29. How many days work at $1.25 a day must be given for 6 cords of wood, worth $4.12| a cord? 30 What will 36.48 yards of cloth cost, if 14.25 yards cost $21. 375? 31. If you can buy 13.25/6. of coffee for $2.50, how much can you buy for $325.50 ? DECIMAL FRACTIONS. 197 212. To change a common to a decimal fraction. The value of a fraction is the quotient of the numerate! divided by the denominator (Art. 148). 1. Reduce J to a decimal. If we place a decimal point after the 5, and then OPERATION. write any number of O's, after it, the value of the 8)5.000 numerator will not be changed (Art. 205). T'9f\ If, then, we divide by the denominator, the quo- tient will be the decimal number : hence, RULE. Annex decimal ciphers to the numerator, and then divide by the denominator, pointing off as in division of decimals. 1 . tteduce EXAMPLES. to its equivalent decimal. We here use two ciphers, and therefore point off two decimal places in the quotient, Reduce the following" fractions to decimals OPERATION. 125)635(5.08 625 1000 1000 to a decimal. 1. Reduce -^ to a decimal. 2. Reduce -J-f- to a decimal. 3. Reduce -fa to a decimal. 4. Reduce J and t ^ 5 . 5. Reduce ^5-, f f , and 6. Reduce J and V. Reduce 8. Reduce f, 9. Reduce to a decimal. 213. A decimal fraction may be changed to the form of a vulgar fraction by simply writing its denominator (Art. 202). 212. How do you change a vulgar to a decimal fraction ? 213. How do you change a decimal to the form of a vulgar fraction ? 10. Reduce 11. Reduce 12. Reduce 13. Reduce 14. Reduce T 15. Reduce 16. Reduce 17. Reduce 18. Reduce 198 DENOMINATE DECIMALS. EXAMPLES. 1. What vulgar fraction is equal to .04 ? 2. What vulgar fraction is equal to 3.067 ? 3. What vulgar fraction is equal to 8.275 ? 4. What vulgar fraction is equal to .00049 ? DENOMINATE DECIMALS. 214. A denominate decimal is one in which the unit of the fraction is a denominate number. Thus, .5 of a pound, .6 of a shilling, .7 of a yard, &c., are denominate decimals, in which the units are 1 pound, 1 shilling, 1 yard. CASE I. 215. To change a denominate number to a denominate decimal. 1. Change 9tf. to the decimal of a . ANALYSIS. The denominate unit of the frac- OPERATION. tion is l=24Qd. Then divide Qd. by 240: 2Qd.=l the quotient, .0375 of a pound is the value of 240) 9 (.03 7 5 9dJ. in the decimal of a : hence, ^ . ^ 0375 RULE. Reduce the unit of the required fraction to the unit of the given denominate number, and then divide the denomi- nate number by the result, and the quotient will be the decimal. EXAMPLES. 1. Reduce 7 drams to the decimal of a Ib. avoirdupois. 2. Reduce 26d. to the decimal of a . 3. Reduce .056 poles to the decimal of an acre. 4. Reduce 14 minutes to the decimal of a day. 5. Reduce 21 pints to the decimal of a peck. 6. Reduce 3 hours to the decimal of a day. 7. Reduce 375678 feet to the decimal of a mile. 8. Reduce 36 yards to the decimal of a rod. 9. Reduce .5 quarts to the decimal of a barrel. 10. Reduce .7 of an ounce, avoirdupois, to the decimal of a hundred. 214. What is a denominate decimal ? 215. How do you change a denominate number to a denominate decimal ? DENOMINATE DECIMALS. 199 CASE II. 216. To find the value of a decimal in integers of a less denomination. 1. Find the value of .890.625 bushels. OPERATION. ANALYSIS. Multiplying the decimal by 4, (since 4 890625 pecks make a bushel), we have 3,5625 pecks. Mul- \ tiplying the new decimal by 8, (since 8 quarts make __ _ a peck), we have 4.5 quarts. Then, multiplying 3.562500 this last decimal by 2, (since 2 pints make a quart), 8 we have 1 pint; hence, 4.500000 2 _ . Bpk. Iqts. Ipt. 1.000000 RULE. I. Multiply the decimal by that number which will reduce it to the next less denomination, pointing off as in multiplication of decimal fractions. 1 1 . Multiply the decimal pa rt of the product as before ; and so continue to do until the decimal is reduced to the required denominations. The integers at the left form the answer. EXAMPLES. 1. What is the value of .002084/6. Troy? 2. What is the value of . 625 of a cwt. ? 3. What is the value of . 625 of a gallon ? 4. What is the value of . 3375 ? 5. What is the value of . 3375 of a ton ? 6. What is the value of . 05 of an acre ? 7. What is the value of . 875 pipes of wine ? 8. What is the value of .125 hogshead of beer ? 9. What is the value of . 375 of a year of 365 days ? 10. What is the value of . 085 of a ? 11. What is the value of .86 of a cwt. ? 12. From .82 of a day take .32 of an hour. 13. What is the value of 1.089 miles? 14. What is the value of .09375 of a pound, avoirdupois ? 15. What is the value of .28493 of a year of 365 days ? 16. What is the value of 1.046? 17. What is the value of 1.88 ? 216. How do you find the value of a decimal in integers of a less denomination ? 200 DENOMINATE DECIMALS CASE III. 217. To reduce a compound denominate number to a decimal or mixed number. 1. Reduce 1 4s. 9|c?. to the decimal of a . ANALYSIS. Reducing the f<f. to a decimal (Art. 215), and annexing the result to the 9d, * ? _ I ^ we have 9.75d. Dividing 9 .75d. by 12, (since $T~~ ' 12 pence Is.), and annexing the quotient to y$d. = $ .*l5d. the 4s. we have 4.8125s. Then, dividing by 20 12)9 75c? (since 20s.=l,) and annexing the quotient to the 1, we have 1.240625 : Ans. 1 4s. 9|d. = 1.240625. RULE. Divide the lowest denomination by as many units as make a unit of the next higher, and annex the quotient as a decimal to that higher: then divide as before, and so continue to do until the decimal is reduced to the required denomination. EXAMPLES. 1. Reduce kwk. $da. 5/ir. 30m. 45s, to the denomination of a week. 2. Reduce 2/6. 5oz. I2pwt. Iftgr., to the denomination of a pound. 3. Reduce 3 feet 9 inches to the denomination of yards. 4. Reduce 1/6. 12dr., avoirdupois, to the denomination of pounds. 5. Reduce 5 leagues 2 furlongs to the denomination of leagues. 6. Reduce 46u. %pk. 4=qt. Ipt. to the denomination of bushels. 7. Reduce 5oz. ISpwt. I2gr. to the decimal of a pound. 8. Reduce Ibcwt. 3qr. 2J/6. to the decimal of a ton. 9. Reduce 5A 3/?. 21sg. rd. to the denomination of acres. 10. Reduce 11 pounds to the decimal of a ton. 1 1. Reduce 3efa. l%%xcc. to the decimal of a week. 12. Reduce 146w. 3%qt. to the decimal of a chaldron. 13. Reduce 7m. 7/wr. Ir. to the denomination of miles. 217. How do you reduce a compound denominate number to a decimal V ANALYSIS. 201 ANALYSIS. 218. An analysis of a proposition is an examination of its separate parts, and their connections with each other. The solution of a question, by analysis, consists in an exami- nation of its elements and of the relations which exist between these elements. We determine the elements and the rela- tions which exist between them, in each case, by examining the nature of the question. In analyzing, we reason from a given number to its unit, and then from this unit to the required number. EXAMPLES. 1. If 9 bushels of wheat cost 18 dollars, what will 21 bushels cost ? ANALYSIS. One bushel of wheat will cost one ninth as much as 9 bushels. Since 9 bushels cost 18 dollars, 1 bushel will cost ^ of 18 dollars, or 2 dollars; 27 bushels will cost 27 times as much as 1 bushel : that is, 27 times ^ of 18 dollars or 54 dollars. OPERATION. 18 O>7 V <** =$54 ; Or, ' | 54 .4ns. NOTE. 1. We indicate the operations to be performed, and then cancel the equal factors (Art. 141). 219. Although the currency of the United States is ex- pressed in dollars cents and mills, still in most of the States the dollar (always valued at 100 cents), is reckoned in shil- lings and pence ; thus, In the New England States, in Indiana, Illinois, Missouri, Vir ginia. Kentucky, Tennessee, Mississippi and Texas, the dollar is reckoned at G shillings: In New York, Ohio and Michigan, at 8 shillings: In New Jersey, Pennsylvania, Delaware and Mary land, at 7s. 6d. : In South Carolina, and Georgia, at 4s. 8d. : In Canada and Nova Scotia, at 5 shillings. 21S. What is an analysis ? In what does the solution of a question by analysis consist ? How do we determine the elements and their relations ? How do we reason in analyzing V 202 ANALYSIS. NOTE In many of the States the retail price of articles is given in shillings and pence, and the result, or cost, required in dollars and cents. 2. What will 12 yards of cloth cost, at 5 shillings a yard, New York currency ? ANALYSIS. Since 1 yard cost 5 shillings 12 yards will cost 12 times 5 shillings, or 60 shillings and as 8 shillings make 1 dollar, New York currency, there will be as many dollars as 8 is contain- ed timesin60=$7ir. OPERATION. 5xl2-^8=$7.50; Or, n 5 2 | 15 = ^=$7.50. $!.50. NOTE. The fractional part of a dollar may always be reduced to cents and mills by annexing two or three ciphers to the nume- rator and dividing by the denominator ; or, which is more conve- nient in practice, annex the ciphers to the dividend and continue the division. 3. What will be the cost of 56 bushels of oats at 3s Zd a bushel, New York currency ? OPERATION. Or, 4 | 91 $22.75 Am. NOTE. When the pence is an aliquot part of a shilling the price may be reduced to an improper fraction, which will be the multiplier: thus, 8l 8d.=8i*.= 1 /. Or: the shillings and pence may be reduced to pence; thus, 3s 3d. ~39rf., in which case the, product will be pence, and must be divided by 96, the number of pence in 1 dollar : hence, 220. To find the cost of articles in dollars and cents. 219; In what is the currency of the States expressed ? the currency of the States often reckoned ? 220. How do you find the cost of a commodity ? In what is ANALYSIS. 203 Multiply the commodity by the price and divide theprodutc by the value of a dollar reduced to the same denominational unit. 4. What will 18 yards of satinet cost at 3s. d. a yard, Pennsylvania currency ? OPERATION. Or, * 00 \ $y. | $9 Ans. NOTE. The above rule will apply to the currency in any of the States. In the last example the multiplier is 3s. 9c?.=3J*. =J*. or 46d. The divisor is 7*. W.=7|*.=^f.=90tl, 5. What will 7J/6. of tea cost at 6s. Sd. a pound, New Englan4 currency ? OPERATION. t t$ L 5 **n *}* 3* 20 l Or, 00 3 25 6. What will be the cost of 120?/^s. of cotton cloth at Is. f)d. a yard, Georgia currency ? 7. What will be the cost in New York currency ? 8. What will be the cost in New England currency ? 9. What will be the cost of 75 bushels of potatoes at 3s. 6d., New York currency ? 10. What will it cost to build 148 feet of wall at Is. Sd. per foot, N. Y. currency ? 11. What will a load of wheat, containing 46 J bushels come to at 10s. Sd. a bushel, N. Y. currency? 12. What will 7 yards of Irish linen cost at 3s. 4d. a yard, Pcnn. currency ? 13. Kow many pounds of butter at Is. 4d. a pound must be given for 12 gallons of molasses at 2s. Sd. a gallon ? 204 ANALYSIS. 12 OPERATION. Or, 12 24/6. | 24/6. NOTE. The same rule applies in the last example as in the preceding ones, except that the divisor is the price of the article received in payment, reduced to the same unit as the price of the article bought. 14. What will be the cost of 12cwt. of sugar at 9cZ. per /&. N. Y. currency? OPERATION. 25 9 2 225 NOTE. Reduce the cicts. to Ibs. by multiplying by 4 and then by 25. Then 2 ^ multiply by the price per pound, and then divide by the value of a dollar in the required currency, reduced to the same denomination asjthe price. Ans. $112,50 15. What will be the cost of 9 hogsheads of molasses at Is. 3d. per quart, N. E. currency ? 16. How many days work at 7s. 6c?. a day must be given for 1 2 bushels of apples at 3s. $d. a bushel ? 17. Farmer A exchanged 35 bushels of barley, worth 6s. 4d. t with farmer B for rye worth 7 shillings a bushel : how many bushels of rye did farmer A receive ? 18. Bought the following bill of goods of Mr. Merchant : what did the whole amount to, N. Y. currency ? 12| yards of cambric at Is. 8 " ribbon 21 " calico 6 " alpaca 4 gallons molasses 2J pounds tea 30 " sugar 19. Iff of a yard of cloth cost $3.20, what will -}- of a yard cost ? ANALYSIS. Since 5 eighths of a yard of cloth costs $3,20, 1 eighth of a yard will cost i of $3,20 ; and 1 yard, or 8 eighths, will cost 8 times as muck, or of $3,20, |$ of a yard will cost i as much as 1 yard, or i$ of of $3.20= $4.80. 4d per yard. 2s. 6d. " Is. 3d. " 5s. Qd, " 3s. bd. per gallon. 6s. 6c?. per pound. ANALYSIS. 205 OPERATION. 1.60 , * yi *.20xlx?xi?=$4.SO. Or, & 1 40 $4.80. 20. If 3 j pounds of tea cost 3^ dollars, what will 9 pounds cost? NOTE. Reduce the mixed numbers to improper fractions, and then apply the same mode of reasoning as in the preceding ex- ample. 21. What will 8| cords of wood cost, if 2f cords cost 7J- dollars ? 22. If 6 men can build a boat in 120 days, how long will it take 24 men to build it ? ANALYSIS. Since 6 men can build .a boat in 120 days, it will take 1 man 6 times 120 days, or 720 days, and 24 men can build it in fa of the time that 1 man will require to build it, or fa of G times 120, which is 30 OPERATION. 30 120x6 -=-24 = 30 days. Or, M Ans. 30 days, 23 If 7 men can dig a ditch in 21 days, how many men will be required to dig it in 3 days ? 24. In what time will 12 horses consume a bin of oats, that will last 21 horses 6f weeks ? 25. A merchant bought a number of bales of velvet, each containing 129^ yards, at the rate of 7 dollars for 5 yards, and sold them at the rate of 1 1 dollars for 7 yards ; and gained 200 dollars by the bargain : how many bales were there ? ANALYSTS Since he paid 7 dollars for 5 yards, for 1 yard he paid ^ of $7 or I of 1 dollar ; and since he received 11 dollars for 7 yards, for 1 yard he received | of 11 dollars or V- of 1 dollar He gained on 1 yard the difference between and V~= - 3 5 r of a dol lar. Since his whole gain was 200 dollars, he had as many yards as the gain on one yard is contained times in his whole gain, or as :ft, is contained times in 200. And there were as many bales as 129 1^, (the number of yards in one bale), is contained times in the whole number of yards ^^ ; which gives 9 bales. 206 ANALYSIS. OPERATION. = 3500, number of yards in a bale : * <& -=-^ 6 5=- 2 -% - -, whole number of yards: ^00 LAO_0 -9 K~l $$00 200 * 26. Suppose a number of bales of cloth, each containing 133^ yards, to be bought at the rate of 12 yards for 11 dol- lars, and sold at the rate of 8 yards for 7 dollars, and the loss in trade to be $100 : how many bales are there ? 27. If a piece of cloth 9 feet long and 3 feet wide, contain 3 square yards ; how long must a piece of cloth that is 2f feet wide be, to contain the same number of yards ? 28. A can mow an acre of grass in 4 hours, B in 6 hours, and C in 8 hours. How many days, working 9 hours a day, would they require to mow 39 acres ? ANALYSIS. Since A can mow an acre in 4 hours, B in 6 hours, and C in 8 hours, A can mow ^ of an acre, B ^ of an acre, and C ^ of an acre in 1 hour. Together they can mow i-ri+|=H of an acre in 1 hour. And since they can mow 13 twenty-fourths of an acre in 1 hour, they can mow 1 twenty fourth of an acre in ^ of 1 hour ; and 1 acre, or f^, in 24 times -jV ^f,- of 1 hour and to mow 39 acres, they will require 39 times ^ ^ hours, which reduced to days of 9 hours each, gives 8 days. OPERATION. l-H+!=Mhours. 8 $ n v* x yX0 = 8 days. Or, $ $ Am. \ 8 days. 29. A can do a piece of work in 4 days, and B can do the same in 6 days ; in what time can they both do the work if they labor together ? 30. If 6 men can do a piece of work in 10 days, how long will it take 5 men to do it? ANALYSIS. If G men can do a piece of work in 10 days, 1 man will require 6 times as long, or 60 days to do the same work Five men will require but one fifth as long as one man or 60^-5 =-12 days. ANALYSIS 207 OPERATION. 10x6-^5=12 days. 6 Ans. | 12 days. 31. Three men together can perform a piece of work in 9 days. A alone can do it in 18 days, B in 27 days ; in what time can C do it alone ? 32. A and B can build a wall on one side of a square piece of ground in 3 days ; A and C in 4 days ; B and C in 6 days : what time will they require, working together, to complete the wall enclosing the square ? 33. Three men hire a pasture, for which they pay 66 dol- lars. The first puts in 2 horses 3 weeks ; the second 6 horses for 2J weeks; the third 9 horses for 1J weeks: how much ought eaeh to pay ? ANALYSIS. The pasturage of 2 horses for 3 weeks, would he the same as the pasturage of 1 horse 2 times 3 weeks, or 6 weeks ; that of six horses 2^ weeks, the same as for 1 horse 6 times 2 weeks, or 15 weeks ; and that of 9 horses 1^ weeks, the same as 1 horse for 9 times H weeks, or 12 weeks. The three persons had an equivalent for the pasturage of 1 horse for 6+15-f 12 -33 weeks ; therefore, the first must pay ^j, the second i, and the third 41 of 66 dollars OPERATION. 3 x2=6; then $66x T r =$12. 1st 21x6=15; " $66 x J$ =$30. 2d. Ijx9 = 12; " $66 x if =$24. 3d. 34. Two persons, A and B, cuter into partnership, and gain $175. A puts in 75 dollars for 4 months, and B puts in 100 dollars for 6 months : what is each one's share of the gain ? 35. Three men engage to build a house for 580 dollars. The first one employed 4 hands, the second 5 hands, and the third 7 hands. The first man's hands worked three times as many days as the third, and the second man's hands twice as many days as the third man's hands : how much must each receive ? 208 ANALYSIS. 36. If 8 students spend $192 in 6 months, how much will 12 students spend in 20 months ? ANALYSIS. Since 8 students spend $192, one student will spend i of $192, in 6 months , in 1 month 1 student will spend -^ of of $192- $4. Twelve students will spend, in 1 month, 12 times as much as 1 student, and in 20 months they will spend 20 times as much as in 1 month. OPERATION. 24 2 -w i i n 20 XXTXyXY=$960. 48 20 $960. Ans. 31 If 6 men can build a wall 80 feet long, 6 feet wide, and 4 feet high, in 15 days, in what time can 18 men build one 240 feet long, 8 feet wide, and 6 feet high ? ANALYSIS. Since it takes 6 men 15 days to build a wall, it will take 1 man 6 times 15 days, or 90 days, to build the same wall. To build a wall 1 foot long, will require - 8 \ r as long as to build one 80 feet long ; to build one 1 foot wide, i as long as to build one 4 feet wide ; and to build one 1 foot high, as long as to build one 6 feet high, 18 men can build the same wall in ^ of the time that one man can build it : but to build one 240 feet long, will take them 240 times as long as to build one 1 foot in length ; to build one 8 feet wide, 8 limes as long as to build one 1 foot wide, and to build one C feet high, 6 times as long as to build one 1 foot high. OPERATION. $ 2 15x0 1 1 1 1 &<0 $ $0 ~~I X $0 X >I X X ;F$ X ~T x ;r x I ^ * *,! 15 Ans. i 30 days. 38 If 96/6s. of bread be sufficient to serve 5 men 12 days, how many days will 57/6. serve 19 men? ANALYSIS. 209 39. If a man travel 220 miles in 10 days, travelling 12 hours a day, in how many days will he travel 880 miles, travelling 16 hours a day? 40. If a family of 12 persons consume a certain quantity of provisions in 6 days, how long will the same provisions last a family of 8 persons ? 41. If 9 men pay $135 for 5 weeks' board, how much must 8 men pay for 4 weeks' board ? 42. If 10 bushels of wheat are equal to 40 bushels of corn, and 28 bushels of corn to 56 pounds of butter, and 39 pounds of butter to 1 cord of wood ; how much wheat is 12 cords of wood worth ? ANALYSIS. Since 10 bushels of wheat are worth 40 bushels of corn, 1 bushel of corn is worth > of 10 bushels of wheat, or i of a bushel ; 28 bushels are worth 28 times of a bushel of wheat, or 7 bushels : since 28 bushels of corn, or 7 bushels of wheat are*worth 56 pounds of butter, 1 pound of butter is worth ^g of 7=i of a bushel of wheat, and 39 pounds are worth 39 times as much as 1 pound, or 39*^=^ bushels of wheat; and since 39 pounds of butter, or ^ bushels of wheat are worth 1 cord of wood, 12 cords are worth 12 times as much, or 12x^=58 bushels. OPERATION. 3 ro i n i 39 xt V rf A />> -| V v i 2 39 o n 3 117=5816^. NOTE. Always commence analysing from the term which is of the same name or kind as the required answer. 43. If 35 women can do as much work as 20 boys, and 16 boys can do as much as 7 men : how many women can do the work of 18 men ? 44. If 36 shillings in New York, are equal to 27 shillings in Massachusetts, and 24 shillings in Massachusetts are equal to 30 shillings in Pennsylvania, and 45 shillings in Pennsyl- vania are equal to 28 shillings in Georgia ; how many shil- lings in Georgia are equal to 72 shillings in New York ? 14 210 PROMISCUOUS EXAMPLES PROMISCUOUS EXAMPLES IN ANALYSIS. 1. How many sheep at 4 dollars a head must I give for 6 cows, worth 12 dollars apiece ? 2. If 7 yards of cloth cost $49, what will 16 yards cost ? 3. If 36 men can build a house in 16 days, how long will it take 12 men to build it? 4. If 3 pounds of butter cost 7J shillings, what will 12 pounds cost ? 5. If 5 1 bushels of potatoes cost $2f, how much will 12 J bushels cost ? 6. How many barrels of apples, worth 1 2 shillings a barrel, will pay for 16 yards of cloth, worth 9s. Qd. a yard ? 7. If 31 J gallons of molasses are worth $9f , what are 5J gallons worth ? 8. What is the value of 24| bushels of corn, at 5s. *ld. a bushel, New York currency ? 9. How much rye, at 8s. Zd. per bushel, must be given for 40 gallons of whisky, worth 2s. 9d. a gallon? 10. If it take 44 yards of carpeting, that is 1 J yards wide, to cover a floor, how many yards of yards wide, will it take to cover the same floor ? 11. If a piece of wall paper, 14 yards long and 1J feet wide, will cover a certain piece of wall, how long must an- other piece be, that is 2 feet wide, to cover the same wall ? 12. If 5 men spend $200 in 160 days, how long will $300 last 12 men at the same rate ? 13. If 1 acre of land cost of f of of $50, what will 3| acres cost ? 14. Three carpenters can finish a house in 2 months ; two of them can do it in 2J months : how long will it take the third to do it alone ? 15. Three persons bought 2 barrels of flour for 15 dollars. The first one ate from them 2 months, the second 3 months, and the third 7 months : how much should each pay ? 16. What quantity of beer will serve 4 persons 18| days, if 6 persons drink 7 gallons in 4 days ? IN ANALYSIS. 211 17. If 9 persons use If pounds of tea in a month, how much will 10 persons use in a year ? 18. If | of f of a gallon of wine cost f of a dollar, what will 5 J gallons cost ? 19. How many yards of carpeting, 1| yards wide, will it take to cover a floor that is 4f yards wide and 6 and three- fifths yards long ? 20. Three persons bought a hogshead of sugar containing 413 pounds. The first paid $2J as often as the second paid $3 J, and as often as the third paid $4 : what was each one's share of the sugar ? 21. A, with the assistance of B, can build a wall 2 feet wide, 3 feet high, and 30 feet long, in 4 days ; but with the assistance of C, they can do it in 3 1 days : in how many days can C do it alone ? 22. If two persons engage in a business, where one advances $875, aritt the other $625, and they gain $300, what is each one's share. 23. A person purchased f of a vessel, and divided it into 5 equal shares, and sold each of those shares for $1200 : what was the value of the whole vessel ? 24. How many yards of paper, f of a yard wide, will be sufficient to paper a room 10 yards square and 3 yards high ? 25. What will be the cost of 45#>s. of coffee, New Jersey currency, if 9?6s. cost 27 shillings ? 26. What will be the cost of 3 barrels of sugar, each weigh- ing %cwt. at 10c?. per pound, Illinois currency? 27. If 12 men reap 80 acres in 6 days, in how many days will 25 men reap 200 acres ? 28. If 4 men are paid 24 dollars for 3 days' labor, how many men may be employed 16 days for $96 ? 29. If $25 will supply a family with flour at $7.50 a bar- rel for 2 months, how long would $45 last the same family when flour is worth $6.75 per barrel ? 30. A wall to be built to the height of 27 feet, was raised to the height of 9 feet by 1 2 men in 6 days : how many men must be employed to finish the wall in 4 days at the same rate of working ? 212 PROMISCUOUS EXAMPLES. 31. A, B and C, sent a drove of hogs to market, of which A owned 105, B 75, and C 120. On the way 60 died : how many must each lose ? 32. Three men, A, B and C, agree to do a piece of work, for which they are to receive $315. A works 8 days, 10 J hours a day ; B 9 j days, 8 hours a day ; and C, 4 days, 12 hours a day : what is each one's share ? 33. If 1 barrels of apples will pay for 5 cords of wood, and 12 cords of wood for 4 tons of hay, how many barrels of apples will pay for 9 tons of hay ? 34. Out of a cistern that is f full is drawn 140 gallons, when it is found to be \ full : how much does it hold ? 35. If .7 of a gallon of wine cost $2.25, what will .25 of a gallon cost ? 36. If it take 5.1 yards of cloth, 1.25 yards wide, to make a gentleman's cloak, how much surge, f yards wide, will be required to line it ? 37. A and B have the same income. A saves | of his annually ; but B, by spending $200 a year more than A, at the end of 5 years find himself $160 in debt : what is their income ? 38. A father gave his younger son $420, which was | of what he gave to his elder s.on ; and 3 times the elder son's portion was \ the value of the father's estate : what was the value of the estate ? 39. Divide $176.40 among 3 persons, so that the first shall have twice as much as the second, and the third three times as much as the first : what is each one's share ? 40. A gentleman having a purse of money, gave \ of it for a span of horses ; of of the remainder for a carriage : when he found that he had but $100 left : how much was in his purse before any was taken out ? 41. A merchant tailor bought a number of pieces of cloth, each containing 25^ yards, at the rate of 3 yards for 4 dol- lars, and sold them at the rate of 5 yards for 13 dollars, and gained by the operation 96 dollars : how many pieces did he buy? RATIO AND PROPORTION. 213 RATIO AND PROPORTION. 221. Two numbers having the same unit, may be com- pared in two ways : 1st. By considering how much one is greater or less than the other, which is shown by their difference ; and, 3d. By considering how many times one is contained in the other, which is shown by their quotient. In comparing two numbers, one with the other, by means of their difference, the less is always taken from the greater. In comparing two numbers, one with the other, by means of their quotient, one of them must be regarded as a standard which measures the other, and the quotient which arises by dividing by the standard, is called the ratio. 222. Every ratio is derived from two numbers : the first is called the antecedent, and the second the consequent: each is called a term, and the two, taken together, are called a couplet. The antecedent will be regarded as the standard. If the numbers 3 and 12 be compared by their difference, the result of the comparison will be 9 ; for, 12 exceeds 3 by 9. If they are compared by means of their quotient, the result will be 4 ; for, 3 is contained in 12, 4 tunes : that is, 3 measuring 12, gives 4. 223. The ratio of one number to another is expressed in two ways : 1st. By a colon ; thus, 3 : 12 ; and is read, 3 is to 12 ; or, o measuring 12. 12 2d. In a fractional form, as; or, 3 measuring 12. 231. In how many ways may two numbers, having the same unit, be compared with each other ? If you compare by their difference, how do you find it ? If you compare by the quotient, how do you regard one of the numbers ? What is the ratio ? 222. From how many terms is a ratio derived ? What is the first term called ? What is the second called ? Which is the standard ? 2~53. How may the ratio of two numbers be expressed ? How read ? 214 RATIO AND PROPORTION. 224. If two couplets have the same ratio, their terms are said to be proportional : the couplets 3 : 12 and 1 : 4 have the same ratio 4 ; hence, the terms arc proportional, and are written, 3 : 12 : : 1 : 4 by simply placing a double colon between the couplets. The terms are read 3 is to 12 as 1 is to 4, and taken together, they are called a proportion : hence, A proportion is a comparison of the terms of two equal ratios* 224. If two couplets have the same ratio, what is said of the terms ? How are they written V How read ? What is a proportion ? * Some authors, of high authority, make the consequent the stand- ard and divide the antecedent by it to determine the ratio of the couplet. The ratio 3 : 13 is the same as that of 1:4 by both methods ; for, if the antecedent be made the standard, the ratio is 4 ; if the conse- quent be made the standard, the ratio is one-fourth. The question is, which method should be adopted V The unit 1 is the number from which all other numbers are derived, and by which they are measured. The question is, how do we most readily apprehend and express the relation between 1 and 4 ? Ask a child, and he will answer, "the dif- ference is 3." But when you ask him, "how many 1's are there in 4V" he will answer, "4," using 1 as the standard. Thus, we begin to teach by using the standard 1 : that is, by dividing 4byl. Now, the relation between 3 and 13 is the same as that between 1 and 4; if then, we divide 4 by 1, we must also divide 13 by 3. Do we, indeed, clearly apprehend the ratio of 3 to 12, until we have referred to 1 as a standard ? Is the mind satisfied until it has clearly perceived that the ratio of 3 to 13 is the same as that of 1 to 4 ? In the Rule of Three we always look for the result in the 4th term. Now, if we wish to find the ratio of 3 to 13, by referring to 1 as a stand- ard, we have 3 : 13 : : 1 : ratio, which brings the result in the right place. But if we define ratio to be the antecedent divided by the consequent, we should have 3 : 12 : : ratio : 1, which would bring the ratio, or required number, in the 3d place, RATIO AND PROPORTION. 215 What are the ratios of the proportions, 3 : 9 : : 12 : 36? 2 : 10 : : 12 : 60? 4 : 2 : : 8 : 4? 9 : 1 : < 90 : 10? 225. The 1st and 4th ter-ms of a proportion are called the extremes : the 2d and 3d terms, the means. Thus, in the pro- portion, 3 : 12 : : 6 : 24 3 and 24 are the extremes, and 12 and 6 the means: 12 24 Since (Art. 224), Y^lp we shall have, by reducing to a common denominator, 12x6_24x3 !Tx~6~ 6x3' But since the fractions are equal, and have the same deno- minators, their numerators must be equal, viz. ; 12x6=24x3; that is, In any proportion, the product of the extremes is equal to the product of the means. Thus, in the proportions, 1 : 6 : : 2 : 12 ; we have 1 x 12= 6x2; 4 : 12 : : S : 24 ; " " 4x24 = 12x8. 220. Since, in any proportion, the product of the extremes is equal to the product of the means, it follows that, In all cases, the numerical value of a quantity is the number of times which that quantity contains an assumed standard, called its unit of If we would find that numerical value, in its right place, we must say, standard : quantity : : 1 : numerical value : but if we take the other method, we have quantity : standard : : numerical value : 1, which brings the numerical value in the wrong place. 216 RATIO AND PROPORTION". 1st. If the product of the means be divided by one of the extremes, the quotient will be the other extreme. Thus, in the proportion 3 : 12 : : 6: 24, we have 3 x 24 = 12 x 6 ; then, if 12, the product of the means, be divided by one of the extremes, 3, the quotient will be the other extreme, 24 : or, if the product be divided by 24, the quotient will be 3. 2d. If the product of the extreme? be divided by either of the means, the quotient ivill be the other mean. Thus, if 3 x 23=12 x 6 = 72 be divided by 12, the quotient will be 6 or if it be divided by 6, the quotient will be 12. EXAMPLES. 1. The first three terms of a proportion are 3, 9 and 12 : what is the fourth term ? 2 The first three terms of a proportion are 4, 16 and 15 : what is the 4th term ? 3. The first, second, and fourth terms of a proportion are 6, 12 and 24 : what is the third term ? 4. The second, third, and fourth terms of a proportion are 9, 6 and 24 : what is the first term ? 5. The first, second and fourth terms are 9, 18 and 48 : what is the third term ? 227. Simple and Compound Eatio. The ratio of two single numbers is called a Simple Eatio, .and the proportion which arises from the equality of two such ratios, a Simple Proportion. 225. Which are the extremes of a proportion ? Which the means ? What is the product of the extremes equal to ? 226. If the product of the means be divided hy one of the extremes, what will the quotient be ? If the product of the means be divided by either extreme, what will the quotient be ? 227. What is a simple ratio ? What is the proportion called which comes from the equality of two simple ratios? What is a compound ratio ? What is a compound proportion ? RATIO AND PROPORTION. 217 If the terms of one ratio be multiplied by the terms of an- other, antecedent by antecedent and consequent by conse- quent, the ratio of the products is called a Compound Ratio- Thus,^if the two ratios 3 : 6 and 4 : 12 be multiplied together, we shall have the compound ratio 3x4 : 6x12, or 12 : 72 ; In which the ratio is equal to the product of the simple ratios. A proportion formed from the equality of two compound ratios, or from the equality of a compound ratio and a simple ratio, is called a Compound Proportion. 228. What part one number is of another. When the standard, or antecedent, is greater than the number which it measures, the ratio is a proper fraction, and is such a part of 1, as the number measured is of the standard. 1. What part of 12 is 3 ? that is, what part of the stand- ard 12, is 3 ? 12 : 3 : : 1 : I; that is, the number measured is one-fourth of the standard. 2. What part of 9 is 2 ? 3. What part of 16 is 4? 4. What part of 100 is 20 ? 5. What part of 300 is 200 ? 6. What part of 36 is 144 ? 7. 3 is what part of 12 ? 8. 5 is what part of 20 ? 9. 8 is what part of 56 ? 10. 9 is what part of 8 ? 11. 12 is what part of 132 ? NOTE. The standard is generally preceded by the word of, and in comparing numbers, may be named second, as in examples 7, 8, 1), 10 and 11, but it must be always be used as a divisor, and should be placed first in the statement. 238. When the standard is greater than the consequent, how may the ratio be compared ? What part is 3 of 1 ? 5 of 1 ? What part is 4 of 2 ? 12 of 3 ? 7 of 5 ? 218 SINGLE RULE OF THREE. SINGLE RULE OF THREE. 229. The Single Rule of Three is an application of the principle of simple ratios. Three numbers are always given aixl a fourth required. The ratio between two of the given numbers is the same as that between the third and the required number. 1. If 3 yards of cloth cost $12, what will 6 yards cost at the same rate ? NOTE. We shall denote the required term of tlie proportion by the letter x. STATEMENT. yd. yd. $ 3 : 6 : : 12 OPERATION. 12 o : x ANALYSIS. The condition, " at the same rate," requires that the quantity 3 yards must have the same ratio to the quantity 6 yards, as $12, the cost of 3 yards, to x dol- lars, the cost of 12 yards. Since the product of the two extremes is equal to the product of the two means, (Art. 235), 3xz=Gxl2; and if 3x^=6x12, x must be equal to this product divided by 3 : A^ C J-AQA that is, The 4th term is equal to the product of the second and third terms divided by the first. 2. If 56 dollars will buy 14 yards of broadcloth, how many yards, at the same rate, can be bought for 84 dollars ? ANALYSIS. Fifty-six dollars, (being the cost of 14 yards of cloth), has the same ratio to $84, as 14 yards has to the number of yards which $84 will buy NOTE. When the vertical line is used, the required term, (which is denoted by a;), is written on the left STATEMENT. $ $ yd. yd. 56 : 84 : : 14 : x OPERATION 21 229. What is the Single Rule of Three ? How many numbers are fivcn ? How many required ? What ratio exists between two of the given numbers ? SINGLE RULE OF THREE. 219 230. Hence, we have the following RULE I. Write the number which is of the same kind with the answer for the third term, the number named in connection with it for the first term, and the remaining number for the second term. II. Multiply the second and third terms together, and divide the product by the first term : Or, Multiply the third term by the ratio of the first and second. NOTES. 1. If the first and second terms have different units, they must be reduced to the same unit. 2. If the third term is a compound denominate number, it must be reduced to its smallest unit. 3. The preparation of the terms, and writing them in their pro- per places, is called the statement. EXAMPLES. 1. If I can walk 84 miles in 3 days, how far can I walk in 11 days? 2. If 4 hats cost $12, what will be the cost of 55 hats at the same rate ? 3. If 40 yards of cloth cost $170, what will 325 yards cost at the same rate ? 4. If 240 sheep produce 660 pounds of wool, how many pounds will be obtained from 1200 sheep? 5 If 2 gallons of molasses cost 65 cents, what will 3 hogs- heads cost ? 6. If a man travels at the rate of 210 miles in 6 days, how far will he travel in a year, supposing him not to travel on Sundays ? 7. If 4 yards of cloth cost $13, what will be the cost of 3 pieces, each containing 25 yards ? 8. If 48 yards of cloth cost $67.25, what will 144 yards cost at the same rate ? 9. If 3 common steps, or paces, are equal to 2 yards, how many yards are there in 1 60 paces ? 10. If 750 men require 22500 rations of bread for a month, how many rations will a garrison of 1200 men require ? 235. Give the rule for the statement. Give the rule for finding the fourth term. 220 SINGLE RULE OF THREE. 11. A cistern containing 200 gallons is filled by a pipe which discharges 3 gallons in 5 minutes ; but the cistern has a leak which empties at the rate of 1 gallon in 5 minutes. If the water begins to run in when the cistern is empty, how long will it run before filling the cistern ? 12. If 14| yards of cloth cost $19*, how much will 19 J yards cost ? NOTE. First make the STATEMENT. statement ; then change tlio yd. yd, $ $ mixed numbers to im- \\ : \C)1 . : IQi : % proper fractions, after which arrange the terms, and cancel equal factors according to previous in- struction. 13. If - of a yard of cloth cost - of a dollar, what will 2 \ yards cost? 14. If y\ of a ship cost 273 2s. Qd., what will ^ of her cost ? 15. If 1 T 4 T bushels of wheat cost $2*, how much will 60 bushels cost ? 16. If 4| yards of cloth cost $9.15, what will 13| yards cost? 17. If a post 8 feet high cast a shadow 12 feet in length, what must be the height of a tree that casts a shadow 122 feet in length, at the same time of day ? 18. If ^cwt. Iqr. of sugar cost $64.96, what will be the cost of kcwt. 2qr. ? 19. A merchant failing in trade, pays 65 cents for every dollar which he owes : he owes A $2750, and B $1975 : how much does he pay each ? 20. If 6 sheep cost $15, and a lamb costs one-third as much as a sheep, what will 27 lambs cost? 21. If 2/6s. of beef cost J of a dollar, what will 30/6*. cost? 22. If 4-J- gallons of molasses cost $2f , how much is it per quart ? 23. A man receives f of his income, and finds it equal to $3724.16 : how much is his whole income ? SINGLE RULE OF THREE. 221 24. If 4 barrels of flour cost $34 f, how much can be bought for $175? 25. If 2 gallons of molasses cost 65 cents, what will 3 hogsheads cost ? 26. What is the cost of -6 bushels of coal at the rate of 1 Us. Qd. a chaldron? 27. What quantity of corn can I buy for 90 guineas, at the rate of 6 shillings a bushel ? 28. A merchant failing in trade owes $3500, and his effects are sold for $2100 : how much does B. receive, to whom he owes $420 ? 29. If 3 yards of broadcloth cost as much as 4 yards of cassimere, how much cassimere can be bought for 18 yards of broadcloth ? 30. If 7 hats cost as much as 25 pair of gloves, worth 84 cents a pair, how many hats can be purchased for $216 ? 31. How many barrels of apples can be bought for $114.33, if 7 barrels cost $21.63? 32. If 27 pounds of butter will buy 45 pounds of sugar, how much butter will buy 36 pounds of sugar ? 33. If 42J tons of coal cost $206.21, what will be the cost of 2J tons ? 34. If 40 gallons run into a cistern, holding 700 gallons, in an hour, and 15 run out, in what time will it be filled ? 35. A piece of land of a certain length and 12 J rods in width, contains 1 J acres, how much would there be in a piece of the same length 26 f rods wide ? 36. If 13 men can be boarded 1 week for $39,585, what will it cost to board 3 men and 6 women the same time, the women being boarded at half price ? 37. What will 75 bushels of wheat cost, if 4 bushels 3 pecks cost $10.687? 38. What will be the cost, in United States money, of 324 yards 3qrs. of cloth, at 5s. d. New York currency, for 2 yards ? 39. At $1.12J a square foot, what will it cost to pave a floor 18 feet long and 12ft. (tin. wide ? 222 CAUSE AND EFFECT. CAUSE AND EFFECT. 231. Whatever produces effects, as men at work, animals eating, time, goods purchased or sold, money lent, and the like, may be regarded as causes. Causes are of two kinds, simple and compound. A simple cause has but a single element, as men at work, a portion of time, goods purchased or sold, and the like. A compound cause is made up of two or more simple ele- ments, such as men at work taken in connection with time, and the like. 232. The results of causes, as work done, provisions con- sumed, money paid, cost of goods, and the like, may be re- garded as effects. A simple effect is one which has but a single element ; a compound effect is one which arises from the multiplication of two or more elements. 233. Causes which are of the same kind, that is, which can be reduced to the same unit, may be compared with each other ; and effects which are of the same kind may likewise be compared with each other. From the nature of causes and effects, we know that 1st Cause : 2d Cause : : 1st Effect : 2d Effect ; and, 1st Effect : 2d Effect : : 1st Cause : 2d Cause. 234. Simple causes and simple effects give rise to simple ratios. Compound causes or compound effects give rise to compound ratios. 331. What arc causes? How many kinds of causes are there? What is a simple cause ? What is a compound cause ? 1 232. What are effects? What is a simple effect? What is a com- pound effect? 233. What causes are of the same kind ? What causes may be com- pared with each other ? What do we infer from the nature of causes and effects ? 234. What gives rise to simple ratios ? DOUBLE RULE OF THREE. 223 DOUBLE RULE OF THREE. 236. The Double Rule of Three is an application of the principles of compound proportion. It embraces all that class of questions in which the causes are compound, or in which the effects are compound ; arid is divided into two parts : 1st When the compound causes produce the same effects ; 2<2. When the compound causes produce different effects. 237. When the compound causes produce the same effects. 1. If 6 men can dig a ditch in 40 days, what time will 30 men require to dig the same ? ANALYSIS. The first cause STATEMENT. men. men. is compounded of 6 men, and ' . on 40 days, the time required to : OIJ do the work, and n equal to days. days. what 1 man would do in 40 : x G x 40=240 days. 240 : 30 xx The second cause is com- pounded of 30 men and the number of days necessary to #0 do th'} same work, viz : x ditch, ditch. : 1 : i But since the effects are the x ~ 8 davs - same, viz : the work done, the causes must be equal ; hence, the products of the elements of the causes are equal. Therefore, in the solution of all like examples, Write the cause containing the unknown element on the left of the vertical line for a divisor, and the other cause on the right for a dividend. NOTE. This class of questions has generally been arranged under the head of " Rule of Three Inverse." EXAMPLES. 1. A certain work can be done in 12 days, by working 4 hours a day : how many days would it require the same number of men to do the same work, if they worked 6 hours a day? 336. What is the double Rule of Three ? What class of questions does it embrace ? Into how many parts is it divided ? What are they ? 337. What is the rule when the effects are equal ? Under what rule has this class of cases been arranged ? 224: DOUBLE RULE OF THREE. 2. A pasture of a certain extent supplies 30 horses for 18 days : how long will the same pasture supply 20 horses ? 3. If a certain quantity of food will subsist a family of 12 persons 48 days, how long will the same food subsist a family of 8 persons ? 4. If 30 barrels of flour will subsist 100 men for 40 days, how long will it subsist 25 men ? 5. If 90 bushels of oats will feed 40 horses for six days, how many horses would consume the same in 1 2 days ? 6. If a man perform a journey of 22 J days, when the days are 12 hours long, how many days will it take him to per- form the same journey when the days are 15 hours long? 7. If a person drinks 20 bottles of wine per month when it costs 2s. per bottle, how much must he drink without increas- ing the expense when it costs 2s. 6e?. per bottle ? 8. If 9 men in 18 days will cut 150 acres of grass, how many men will cut the same in 27 days ? 9. If a garrison of 536 men have provisions for 326 days, how long will those provisions last if the garrison be increased to 1304 men ? 10. A pasture of a certain extent having supplied a body of horse, consisting of 3000, with forage for 18 days : how many days would the same pasture have supplied a body of 2000 horse ? 11. What length must be cut off from a board that is 9 inches wide, to make a square foot, that is, as much as is contained in 12 inches in length and 12 in breadth ? 12. If a certain sum of money will buy 40 bushels of oats at 45 cents a bushel, how many bushels of barley will the same money buy at 72 cents a bushel ? 13. If 30 barrels of flour will support 100 men for 40 days, how long would it subsist 400 men ? 14. The governor of a besieged place has provisions for 54 days, at the rate of 2/6. of bread per day, but is desirous of prolonging the siege to 80 days in expectation of succor : what must be the ration of bread ? DOUBLE RULE OF THREE. 225 238. When the Compound Causes produce different Effects. In this class of questions, either a cause, or a single ele- ment of a cause may. be required ; or an effect, or a single element of an effect may be required. 1. If a family of 6 persons expend $300 in 8 months, how much will serve a family of 15 persons for 20 months ? ANALYSIS. In this example the second effect is required ; and the statement may be read thus : If 6 persons in 8 months expend $300, 15 persons in 20 months will expend how many (or x) dollars ? OPERATION 15 5 ( *0 & 25 X #=1875 Ans. STATEMENT. 1st Cause : 2d Cause : : 1st Effect : 2d Effect 15) 20 j Or, 6x8 :. 15x20 $300 300 2. If 16 men, in 12 days, build 18 feet of wall, how many men must be employed to build 72 feet in 8 days ? ANALYSIS. In this example an element of the second cause is required, viz : the number of men. The question may be read thus : If 16 men, in 12 days, build 18 feet of wall, how many (or x) men, in 8 days, will build 72 feet of wall ? . , * $ $ x OPERATION. ^ ,4 " * 9 Jf 12 =96 men. STATEMENT. 1} 1S ^ 1 Q ^79 . io . \ A. in , 12 j Or, 16 x 12 : 3. If 32 men build a wall 36 feet long, 8 feet high, and 4 feet thick, in 4 days, working 12 hours a day how long a wall, that is 6 feet high, and 3 feet thick can 48 men build in 36 days, working 9 hours a day ? 238. When the compound causes produce different effects, what will always be required ? 15 226 DOUBLE BULE OF THKEE. ' OPERATION. ) 48 36) x Y : 36 : : 8> : 6 ) 9' 4) 3 ANALYSIS. In this example an element of the second effect is required, viz : the length of the wall, and the question may be read thus : If 32 men, in 4 days, working 12 hours a day, can build a wall 36 feet long, 8 feet high, and 4 feet thick, 48 men in 36 days, working 9 hours a day, can build a wall how many (or x) feet long, 6 feet high, and 3 feet thick ? #1=648 feet. STATEMENT. 32 4 12 Or, 32x4x12 : 48x36x9 : : 36x8x4 : #x6x3. 239. Hence, we have the following RULE. I. Arrange the terms in the statement so that the causes shall compose one couplet, and the effects the other, putting x in the place of the required element : II. Then if x fall in one of the extremes, make the product of the means a dividend, and the product of the extremes a divisor; but if x fall in one of the means, make the product of the extremes a dividend, and the product of the means a divisor. EXAMPLES. 1. If I pay $24 for the transportation of 96 barrels of flour 200 miles, what must I pay for the transportation of 480 bar- rels 75 miles ? 2. If 12 ounces of wool be sufficient to make 1| yards of cloth 6 quarters wide, what number of pounds will be required to make 450 yards of flannel 4 quarters wide ? 3. What will be the wages of 9 men for 11 days, if the wages of 6 men for 14 days be $84 ? 4. How long would 406 bushels of oats last 7 horses, if 154 bushels serve 14 horses 44 days ? . If a man travel 217 miles in 7 days, travelling 6 hours 7 tfay, how far would he travel in 9 days if he travelled 11 fiours a day ? 939. What is the rule for finding tho unknown part ? DOUBLE SULE OF THREE. 227 6. If 27 men can mow 20 acres of grass in 5$- days, work- ing 3f hours a day, how many acres can 10 men mow in 4| days, by working 8 J hours a day ? 7. How long will it take 5 men to earn $11250, if 25 men can earn $6250 in 2 years ? 8. If 15 weavers, by working 10 hours a day for 10 days, can make 250 yards of cloth, how many must work 9 hours a day for 15 days to make 60 7 J yards? 9. A regiment of 100 men drank 20 dollars' worth of wine at 30 cents a bottle : how many men, drinking at the same rate, will require 1 2 dollars' worth at 25 cents a bottle ? 10. If a footman travel 341 miles in 7^ days, travelling 12 J hours each day, in how many days, travelling 10^ hours a day, will he travel 155 miles? 11. If 25 persons consume 300 bushels of corn in 1 year, how much will 139 persons consume in 8 months, at the same rate ? 12. How much hay will 32 horses eat in 120 days, if 96 horses eat 3J tons in 7| weeks ? 13. If $2. 45 will pay for painting a surface 21 feet long and 13 J feet wide, what length of surface that is lOf feet wide, can be painted for $31.72 ? 14. How many pounds of thread will it require to make 60 yards of 3 quarters wide, if 7 pounds make 14 yards 6 quarters wide ? 15. If 500 copies of a book, containing 210 pages, require 12 reams of paper, how much paper will be required to print 1200 copies of a book of 280 pages? 16. If a cistern 17J feet long, 10 feet wide, and 13 feet deep, hold 546 barrels of water, how many barrels will a cistern 12 feet long, 10 feet wide, and 7 feet deep, contain ? 17. A contractor agreed to build 24 miles of railroad in 8 months, and for this purpose employed 150 men. At the end of 5 months but 10 miles of the road were built : how many more men must be employed to finish the road in the time agreed upon ? 18. If 336 men, in 5 days of 10 hours each, can dig a trench of 5 degrees of hardness, 70 yards long 3 wide and 2 deep : what length of trench of 6 degrees of hardness, 5 yards wide and 3 yards deep, may be dug by 240 men in 9 days of 12 hours each ? 228 PARTNERSHIP. PARTNERSHIP. 240. PARTNERSHIP is the joining together of two or more persons in trade, with an agreement to share the profits or losses. PARTNERS are those who are united together in carrying on business. CAPITAL, is the amount of money or property employed : DIVIDEND is the gain or profit : Loss is the opposite of profit : 241. The Capital or Stock is the cause of the entire profit : Each man's capital is the cause of his profit : The entire profit or loss is the effect of the whole capital : Each man's profit or loss is the effect of his capital : hence, Wliole Stock : Each man's Stock : : Whole profit or loss : Each man's profit or loss. EXAMPLES. 1. A and B buy certain goods amounting to 160 dollars, of which A pays 90 dollars and B, 70 ; they gain 32 dollars by the purchase : what is each one's share ? OPERATION. 160 : 90 : : 32 : A's share ; or, 160 : : 70 : 32 : B's share ; or, 240. What is a partnership ? What are partners ? What is capital or stock ? What is dividend ? What is loss ? 241. What is the cause of the profit? What is the cause of each man's profit? What is the effect of the whole capital ? What is the effect of each man's capital ? What proportion exists between causes and their effects ? What is the rule ? COMPOUND PARTNERSHIP. 229 Hence, the following RULE. As the whole stock is to each man's share, so is the whole gain or loss to each man's share of the guin or loss. EXAMPLES. 1. A and B have a joint stock of $2100, of which A owns $1800 and B $300 ; they gain in a year $1000 : what is each one's share of the profits ? 2. A, B and C fit out a ship for Liverpool. A contributes $3200, B $5000, and C $4500 ; the profits of the voyage amount to $1905 : what is the portion of each ? 3. Mr. Wilson agrees to put in 5 dollars as often as Mr. Jones puts in 7 ; 'after raising their capital in this way, they trade for 1 year and find their profits to be $3600 : what is the share of each ? 4. A. B and C make up a capital of $20,000 ; B and C each contribute twice as much as A ; but A is to receive one- third of the profits for extra services ; at the end of the year they have gained $4000 : what is each to receive ? 5. A, B and C agree to build a railroad and contribute $18000 of capital, of which B pays 2 dollars, and C, 3 dollars as often as A pays 1 dollar ; they lose $2400 by the opera- tion : what is the loss of each ? COMPOUND PARTNERSHIP. 242. When the causes of profit or loss are compound. "When the partners employ their capital for different periods of time, each cause of profit or loss is compound, being made up of the two elements of capital and t^me. The product of these elements, in each particular case, will be the cause of each man's gain or loss ; and their sum will be the cause of the entire gain or loss : hence, to find each share, Multiply each man 1 stock by the time he continued it in trade ; then say, as the sum of the products is to each product, so is the whole gain or loss to each man's share of the gain or 243. "When is the cause of profit or loss compound ? What arc the elements of the compound caus ? What is the rule in this case? 230 COMPOUND PARTNERSHIP. EXAMPLES. 1. A and B entered into partnership. A put in $840 for 4 months, and B, $650 for 6 months ; they gained $363 : what is each one's share ? OPERATION. A, $840x4-3360 B. 650 x 63900 J 3360 : : QPQ f $168 A's. J3900 :: 363: j $195 B's. 2. A puts in trade $550 for 7 months and B puts in $1625 for 8 months ; they make a profit of $337 : what is the share of each ? 3. A and B hires a pasture, for which they agreed to pay $92.50. A pastures 12 horses for 9 weeks and B 11 horses for 7 weeks : what portion must each pay ? 4. Four traders form a company. A puts in $400 for ft months ; B $600 for 7 months ; C $960 for 8 months ; D $1200 for 9 months. In the course of trade they lost $750 ; how much falls to the share of each ? 5. A, B and C contribute to a capital of $15000 in the following manner : every time A puts in 3 dollars B puts in $5 and C, $7. A's capital remains in trade 1 year ; B's If- years ; and C's 2f years ; at the end of the time there is a profit of $15000 : what is the share of each ? 6. A commenced business January 1st, with a capital of $3400. April 1st, he took B into partnership, with a capital of $2600 ; at the expiration of the year they had gained. $750 : what is each one's share of the gain ? 7. James Fuller, John Brown and William Dexter formed a partnership, under the firm of Fuller, Brown & Co., with a capital of $20000 ; of which Fuller furnished $6000, Brown $5000, and Dexter $9000. At the expiration of 4 months, Fuller furnished $20^)0 more ; at the expiration of 6 months, Brown furnished $2500 more ; and at the end of a year Dex- ter withdrew $2000. At the expiration of one year and a half, they found their profits amounted to $5400 : what was each partner's share ? PERCENTAGE. 231 PERCENTAGE. 243. PERCENTAGE is an allowance made by the hundred. The base of percentage, is the number on which the per- centage is reckoned. PER CENT means by the hundred : thus, 1 per cent means 1 for every hundred ; 2 per cent, 2 for every hundred ; 3 per cent, 3 for every hundred, &c. The allowances, 1 per cent, 2 per cent, 3 per cent, &c., are called rates, and may be expressed decimally, as in the following TABLE. 1 per cent is -01 7 per cent is .07 3 per cent is .03 3 per cent is .08 4 per cent is .04" 15 per cent is .15 5 per cent is .05 68 per cent is .68 6 percent is .06 99 per cent is .99 100 per cent is 1. 150 per cent is 1.50 130 per cent is 1.30 200 per cent is 2. . \ per cent is .005 3| per cent is .035 5| per cent is 0575 ALSO, for, 1-0$ is equal to 1 . for, |g is equal to 1.50 for, |$# is equal to 1.30 for, f $ is equal to 2.00 for, T-^-^2 is equal to .005 for, 3J = .03+.005 = .035 for, 5j=.05+.075 = .OT5 EXAMPLES. Write, decimally, 8J per cent ; 9 per cent ; 6| per cent ; 65J per cent ; 205 per cent ; 327 per cent. 244. To find the percentage of any number. 1. What is the percentage of $320, the rate being 5 per cent? 343. What is per centage? What is the base? What does per cent mean ? What do you understand by 3 per cent ? What is the rate, or rate per cent ? 244. How do yon find the percentage of any number ? 232 PERCENTAGE. ANALYSIS. The rate being 5 per cent, is ex- OPERATION. pressed decimally by .05. We are then to take 320 .05 of the base (which is $320) ; this we do by multiplying $320 by .05. Hence, to find the percentage of a number, $16. 00 Ans. Multiply the number by the rate oppressed decimally, and the product will be the percentage. EXAMPLES. 1. What is the percentage of $657, the rate being 4J per cent? OPERATION NOTE. When the rate cannot be .657 reduced to an exact decimal, it is most Q^I convenient to multiply by the fraction, and then by that part of the rate which 219 = | per cent, is expressed in exact decimals. 2628 = 4 per cent. $28.47 = 41 per cent. Find the percentage of the following numbers : 1. 2J per cent of 650 dollars. 2. 3 per cent of 650 yards. 3. 4 per cent of Slbcwl. 4. 6J per cent of $37.50. 5. 5| per cent of 2704 miles. 6. \ per cent of 1000 oxen. 7 2| per cent of $376. 8. 2^ per cent of 860 sheep. 9. 5 per cent of $327.33. 10. 66| per cent of 420 cows. 11. 105 per cent of 850 tons. 12. 116 per cent of 875/6. 13. 241 per cent of $875.12. 14. 37J per cent of $200. 15. 33^ per cent of $687.24. 16. 87J per cent of $400. 17. 62J per cent of $600. 18. 308 per cent of $225.40. 19. A has $852 deposited in the bank, and wishes to draw out 5 per cent of it : how much must he draw for ? 20. A merchant has 1200 barrels of flour : he shipped 64 per cent of it and sold the remainder : how much did he sell? 21. A merchant bought 1200 hogsheads of molasses. On getting it into his store, he found it short 3| per cent : how many hogsheads were wanting ? 1 22. What is the difference between 5| per cent of $800 and 6J per cent of $1050? PERCENTAGE. 233 23. Two men had each $240. One of them spends 14 per cent, and the other 18| per cent : how many dollars more did one spend than the other ? 24. A man has a capital of $12500 : he puts 15 per cent of it in State Stocks : 33 J per cent in Railroad Stocks, and 25 per cent in bonds and mortgages : what per cent has he left, and what is its value ? 25. A farmer raises 850 bushels of wheat : he agrees to sell 18 per cent of it at $1.25 a bushel ; 50 per cent of it at $1.50 a bushel, and the remainder at $1.75 a bushel : how much does he receive in all ? 245. To find the per cent which one number is of another. 1. What per cent of $16 is $4 ? ANALYSIS. The question is, what part of OPERATION. $16 is $4, when expressed in hundreths: JL- 1 .25. The standard is $16 (Art. 228) : hence, the or 25 p er cent, part is -j*g:^ .25; therefore, the per cent is 25 : hence, to find what per cent one number is of another, Divide by the standard or base, and the quotient, reduced to decimals, will express the rate per cent. NOTE. The standard or base, is generally preceded by the word of. EXAMPLES. 1. What per cent of 20 dollars is 5 dollars? 2. Forty dollars is what per cent of eighty dollars ? 3. What per cent of 200 dollars is 80 dollars ? 4. What per cent of 1250 dollars is 250 dollars ? 5. What per cent of 650 dollars is 250 dollars ? 6. Ninety bushels of wheat is what per cent of ISOO&usJi.? 7. Nine yards of cloth is what per cent of 870 yards ? 8. Forty-eight head of cattle are what per cent of a drove of 1600 ? 9. A man has $550, and purchases goods to the amount of $82.75 : what per cent of his money does he expend? 245. How do you find the per cent which one number is of another ? 234 PERCENTAGE. 10. A merchant goes to New York with $1500 ; he first lays out 20 "per cent, after which he expends $660 : what per cent was his last purchase of the money that remained after his first ? 11. Out of a cask containing 300 gallons, 60 gallons are drawn : what per cent is this ? 12. If I pay $698.23 for 3 hogsheads of molasses and sell them for $837.996, how much do I gain per cent on the money laid out ? 13. A man purchased a farm of 75 acres at $42.40 an acre. He afterwards sold the same farm for $3577.50 : what was his gain per cent on the purchase money ? STOCK, COMMISSION AND BROKERAGE. 246. A CORPORATION is a collection of persons authorized by law to do business together. The law which defines their rights and powers is called a Charter. CAPITAL or STOCK is the money paid in to carry on the business of the Corporation, and the individuals so contributing are called Stockholders. This capital is divided into equal parts called Shares, and the written evidences of ownership are called Certificates. 247. When the United States Government, or any of the States, borrows money, an acknowledgment is given to the lender, in the form of a bond, bearing a fixed interest. Such bonds are called United States Stock, or State Stock. The par value of stock is the number of dollars named in each share. The market value is what the stock brings per share when sold for cash. If the market value is above the par value, the stock is said to be at a premium, or above par ; but if the market value is below the par value, it is said to be at a discount, or below par. 346. What is a corporation ? What is a charter? What is capital or stock ? What are shares ? 347. What are United States Stocks? What are State Stocks? What is the par value of a stock ? What is the market value ? If the market is above the par value, what is said of the stock ? If it is below, what is said of the stock ? What is the market value when above par ? What when below ? COMMISSION AND BROKERAGE. 235 Let l=par value of 1 dollar : l+premium= market value of 1 dollar, 'when above par : 1 discount =: market value of 1 dollar when below par. 248. Commission is an allowance made to an agent for buying or selling, or taking charge of property, and is gen- erally reckoned at a certain rate per cent. The commission, for the purchase or sale of goods in the city of New York, varies from 2J to 12 J per cent, and under some circumstances even higher rates are paid. Brokerage is an allowance made to an agent who buys or sells stocks, uncurrent money, or bills of exchange, and is generally reckoned at so much per cent on the par value of the stock. The brokerage, in the city of New York, is gene- rally one-fourth per cent on the par value of the stock. EXAMPLES. 1. What is the commission on $4396 at per 6 cent? OPERATION. NOTE. We here find the commission, as $4396 in simple percentage, by multiplying by the de- Q g cimal which expresses the rate per cent. : Am. $263.76. 2. A factor sells 60 bales of cotton at $425 per bale, and is to receive 2 J per cent commission : how much must he pay over to his principal ? 3. A drover agrees to purchase a drove of cattle and to sell them in New York city for 5 per cent on what he may re- ceive ; he expends in the purchase $4250, and sells them at an advance of 10 per cent : how much is his commission ? 4. A commission merchant sells goods to the amount of $8750, on which he is to be allowed 2 per cent, but in con- sideration of paying the money over before it is due, he is to receive !- per cent additional : how much must he pay over to his principal ? 5. A broken bank has a circulation of $98000 and pur- chases the bills a,t 85 per cent : how much is made by the operation ? 248. What is commission ? What is brokerage ? 236 PERCENTAGE. 6. Merchant A sent to B, a broker, $3825 to be invested in stock ; B is to receive 2 per cent on the amount paid for the stock : what was the value of the stock purchased ? OPERATION. ANALYSTS. Since the broker re- 1 .02)3825 .00($3750vl?is. ceives 2 per cent, it will require 306 $1.02 to purchase 1 dollar's worth of stock; hence, there will be as 765 many dollar's worth purchased as 714 $1.02 is contained times in $3825 ; that is, $3750 worth. 510 7. Mr. Jones sends his broker $18560 to be invested in U. S. Stocks, which are 15 per cent above par ; the broker is to receive one per cent ; how many shares of $100 each can be purchased ? ANALYSIS. Since the premium is 15 per cent, and the brokerage 1 per cent, OPERATION. each dollar of par value will cost $1 1.16)18560 plus the premium, plus the brokerage^ $1.16 : hence, the amount purchased ' quotient, wiU be as many dollars as $1.16 is or, 160 shares, contained times in $18560. 8. I have $4999.89 to be laid out in stocks, which are 15 per cent below par : allowing 2 per cent commission, how much can be purchased at the par value ? ANALYSIS. Since the stock is at a dis- count of 15 per cent, the market value will OPERATION. be 85 per cent ; add 2 per cent, the broker- . 87)4999.89 age, gives 87 per cent=.87. The amount v-^ . ^ . purchased will be as many dollars as .87 is contained times in $4999,89. Hence, to find the amount at par value, Divide the amount to be expended by the market value of $1 plus the brokerage ; and the quotient ivill be the amount in par value. 9. Messrs. Sherman & Co. received of Mr Gilbert $28638.50 to be invested in bank stocks, which are 12i per cent above par, for which they are to receive one-fourth of one per cent commission : how many shares of $127 each can they buy ? LOSS OR GAIN. 237 10. The par value of Illinois Railroad stock is 100. It sells in market at 72 J : if I pay J per cent brokerage, how many shares can I buy for $5820 ? PROFIT AND LOSS. 249. Profit or loss is a process by which merchants dis- cover the amount gained or lost in the purchase and sale of goods. It also instructs them how much to increase or diminish the price of their goods, so as to make or lose so much per cent. EXAMPLES. 1. Bought a piece of cloth containing 75?/d. at $5.25 per yard, and sold it at $5.75 per yard : how much was gained in the trade ? OPERATION. ANALYSIS. We first find tho $5.75 p r i ce of 1 yard, profit on a single yard, and then AC op; oc .^ n f i vnrf i multiply by the number of yards, !^_ co which is *5. 50cfe. profit on 1 yard : then, $0.50x75=$37.50. 2. Bought a piece of calico containing 56 yards, at 27 cents a yard : what must it be sold for per yard to gain $2.24 ? OPERATION. 56 yards at 27 cents=$15.12 ANALYSIS. First find the Profit - 2.24 cost, then add the profit and T , ,, ,, divide the sum by the number Ifc must sel1 f r ' WM. of yards 56)17,36 31 cts. a yard. 250. Knowing the per cent, of gain or loss and the amount received, to find the cost. 1. I sold a parcel of goods for $195.50, on which I made 15 per cent : what did they cost me ? ANALYSIS. 1 dollar of the cost plus 15 per OPERATION. cent, will be what that which cost $1 sold for, 1.15) 195.50 viz , $1.15 : hence, there will be as many ^ K dollars of cost, as $1.15 is contained times in * L1() Ans - what the goods brought. 349. What is loss or gain ? 238 PERCENTAGE. 2. If I sell a parcel of goods for $170, by which I lose 15 per cent, what did they cost ? ANALYSIS. 1 dollar of the cost less 15 per OPERATION. cent, will be what that which cost 1 dollar sold .85) 170 for, viz., $0.85 : hence, there will be as many dollars of cost, as .85 is contained times in what the goods brought. Hence, to find the cost, Divide the amount received by 1 plus the per cent ivhen there is a gain, and by 1 minus the per cent when there is a loss, and the quotient will be the cost. EXAMPLES. 1. Bought a piece of cassimere containing 28 yards at 1 dollars a yard ; but finding it damaged, am willing to sell it at a loss of 15 per cent : how much must be asked per yard? 2. Bought a hogshead of brandy at $1.25 per gallon, and sold it for $78 : was there a loss or gain ? 3. A merchant purchased 3275 bushels of wheat for which he paid $3517.10, but finding it damaged, is willing to lose 10 per cent : what must it sell for per bushel ? 4. Bought a quantity of wine at $1.25 per gallon, but it proves to be bad and am obliged to sell it at 20 per cent less than I gave : how much must I sell it for per gallon ? 5. A farmer sells 125 bushels of corn for 75 cents per bushel ; the purchaser sells it at an advance of 20 per cent : how much did he receive for the corn ? 6. A merchant buys 1 tun of wine for which he pays $725, and wishes to sell it by the hogshead at an advance of 15 per cent : what must be charged per hogshead ? 7. A merchant buys 158 yards of calico for which he pays 20 cents per yard ; one-half is so damaged that he is obliged to sell it at a loss of 6 per cent : the remainder he sells at an advance of 19 per cent : how much did he gain? 8. If I buy coffee at 16 cents and sell it at 20 cents a pound, how much do I make per cent on the money paid ? 250. Knowing the per cent of gain or loss and the amount received how do you find the cost ? INSURANCE. i!39 9. A man bought a house and lot for $1850.50, and sold it for $1517.41 : how much per cent did he lose ? 10. A merchant bought 650 pounds of cheese at 10 cents per pound, and sold it at 12 cents per pound : how much did he gain on the whole, and how much per cent on the money laid out ? 11. Bought cloth at $1.25 per yard, which proving bad, I wish to sell it at a loss of 18 per cent : how much must I ask per yard ? 12. Bought 50 gallons of molasses at 75 cents a gallon, 10 gallons of which leaked out. At what price per gallon must the remainder be sold that I may clear 10 per cent on the cost ? 13. Bought 67 yards of cloth for $112, but 19 yards being spoiled, I am willing to lose 5 per cent : how much must I sell it for per yard ? 14. Bought 67 yards of cloth for $112, but a number of yards being spoiled, I sell the remainder at $2.216| per yard, and lose 5 per cent : how many yards were spoiled ? 15. Bought 2000 bushels of wheat at $1.75 a bushel, from which was manufactured 475 barrels of flour : what must the flour sell for per barrel to gain 25 per cent on the cost of the wheat ? INSURANCE. 251. INSURANCE is an agreement, generally in writing, by which an individual or company bind themselves to exempt the owners of certain property, such as ships, goods, houses, &c., from loss or hazard. The POLICY is the written agreement made by the parties. PREMIUM is the amount paid by him who owns the property to those who insure it, as a compensation for their risk. The premium is generally so much per cent on the property in- sured. EXAMPLES. 1. What would be the premium for the insurance of a house valued at $8754 against loss by fire for one year, at \ per cent ? 251. What is insurance? What is the policy? What is the pre- mium ? How is it reckoned ? PERCENTAGE. 2. What would bo the premium for insuring a ship and cargo, valued at $37500, from New York to Liverpool, at 3 per cent ? 3. What would be the insurance on a ship valued at $47520 at J per cent ; also at J per cent? 4. What would be the insurance on a house valued at $14000 at 1J per cent? 5. What is the insurance on a store and goods valued at $27000, at 2 J per cent ? 6. What is the premium of insurance on $9870 at 14 per cent? 7. A merchant wishes to insure on a vessel and cargo at sea, valued at $28800 : what will be th^ premium at 1| per cent ? 8. A merchant owns three-fourths of a ship valued at $24000, and insures his interest at 2| per cent : what does he pay for his policy ? 9. A merchant learns that his vessel and cargo, valued at $36000, have been injured to the amount of $12000 ; he effects an insurance on the remainder at 5| per cent ; what premium does he pay ? 10. My furniture, worth $3440, is insured at 2f per cent ; my house, worth $1000, at 1 J per cent ; and my barn, horses and carriages, worth $1500, at 3J per cent : what is the whole amount of my insurance ? 11. A man bought a house, and paid the insurance at 2| per cent, the whole of which amounted to $1845 : what was the value of the house and the amount of the insurance ? 12. What would it cost to insure a store, worth $3240, at f per cent, and the stock, worth $7515.75, at f per cent? 13. A merchant imported 250 pieces of broadcloth, each piece containing 36| yards, at $3.25 cents a yard. He paid 4| per cent insurance on the selling price, $4.50 a yard. If the goods were destroyed by fire, and he got the amount of insurance, how much did he make ? 14. A vessel and cargo, worth $65000, are damaged to the amount of 20 per cent, and there is an insurance of 50 per cent on the loss: how much insurance will the owner re- ceive ? INTEREST. 241 INTEREST. 252. INTEREST is an allowance made for the use of money that is borrowed. PRINCIPAL is the money on which interest is paid. AMOUNT is the sum of the Principal and Interest. For example : If I borrow 1 dollar of Mr. Wilson for 1 year, and pay him 7 cents for the use of it ; then, 1 dollar is the principal, 7 cents is the interest, and $1.07 the amount. The RATE of interest is the number of cents paid for the use of 1 dollar for 1 year. Thus, in the above example, th*e rate is 7 per cent per annum. NoTE.-VThe term per cent means, ty the hundred; and per annum means by the year. As interest is always reckoned by the year, the term per annum is understood and omitted. CASE I. 253. To find the interest of any principal for one or more years. 1. What is the interest of $1960 for 4 years, at 7 per cent? ANALYSIS. The rate of interest being 7 per cent, is expressed deci- OPERATION. mallyby.07: hence, each dollar, in $1960 1 year will produce .07 of itself, and A 7 rq fp $1960 will produce .07 of $1960, or $137.20. Therefore, $137.20 is the 137.20 int. for It/r. interest for 1 year, and this interest 4 No. of years, multiplied by 4, gives the interest for AC 4 Q Qft 4 years : hence, the following $D48.U RULE. Multiply the principal by the rate, expressed decimally, and the product by the number of years. 252. What is interest? What is principal? What is amount? What is rate of interest ? \Vhat does per annum mean ? 253. How do you find the interest of any principal for any number of years ? Give the analysis. 242 SIMPLE INTEREST. EXAMPLES. 1. What is the interest of $365.874 for one year, at 5J per cent ? OPERATION. 365.874 ANALYSIS. We first find the in- 951 terest at ^ per cent, and then the - interest at 5 per cent ; the sum is 1.82937 per cent, the interest at 5 per cent. 18.29370 5 per cent. Ans. $20.12307 5J per cent. 2. What is the interest of $650 for one year, at 6 per cent ? 3. What is the interest of $950 for 4 years, at 7 per cent ? 4. What is the amount of $3675 in 3 years, at 7 per cent ? 5. What is the amount of $459 in 5 years, at 8 per cent ? 6. What is the amount of $375 in 2 years, at 7 per cent? 7. What is the interest of $21 1.26 for 1 year, at 4J per ct. ? 8. What is the interest of $1576.91 for 3 years, at 7 per ct. ? 9. What is the amount of $957.08 in 6 years, at 3J per ct. ? 10. What is the interest of $375.45 for 7 years, at 7 per ct. ? 11. What is the amount of $4049.87 in 2 years, at 5 per ct. ? 12. What is the amount of $16199.48 in 16 yrs., at 5J per ct. ? NOTE. When there are years and months, and the months are aliquot parts of a year, multiply the interest for 1 year by the years and months reduced to the fraction of a year. EXAMPLES. 1. What is the interest of $326.50, for 4 years and 2 months, at 7 per cent ? 2. What is the interest of $437.21, for 9 years and 3 months, at 3 per cent ? 3. What is the amount of $1119.48, after 2 years and 6 months, at 7 per cent ? 4. What is the amount of $179.25, after 3 years and 4 months, at 7 per cent? 5. What is the amount of $1046.24, after 4 years and 3 months at 5^ per cent ? SIMPLE INTEREST. 24:3 CASE II. 254. To find the interest on a given principal for any rate and time. 1. What is the interest of $876.48 at 6 per cent, for 4 years 9 months and 14 days ? ANALYSIS. The interest for 1 year is the product of the princi- pal multiplied by the rate If the interest for 1 year be divided by 12, the quotient will be the interest for 1 month : if the interest for 1 month be divided by 30, the quotient will be the interest for 1 day. The interest for 4 years is 4 times the interest for 1 year ; the interest for 9 months, 9 times the interest for 1 month ; and the interest for 14 days, 14 times the interest for 1 day OPERATION. $876.48 .06 12)52.5888=int. for lyr. 52.5888 x 4 =$210.3552 4yr. 30)4.3824 =int. for Imo. 4.3824 x 9 = $ 39.4416 9mo. .14608=int. for Ida. .14608 x 14=$ 2.0451 Udg. Total interest, $251.84194- Hence, we have the following RULE. I. Find the interest for 1 year : II. Divide this interest by 12, and the quotient will be the interest for 1 month : III. Divide the interest for 1 month by 30, and the quo- tient will be the interest for 1 day. IY. Multiply the interest for 1 year by the number of years, the interest for 1 month by the number of months, and the interest for 1 day by the number of days, and the sum of the product will be the required interest. NOTE. In computing interest the month is reckoned at 30 days. 2. What is the interest of $132.26 for 1 year 4 months and 10 days, at 6 per cent per annum ? 3 What is the interest of $25.50 for 1 year 9 months and 12 days, at 6 per cent ? 254. How do you find the interest for any time at any rate ? 244: SIMPLE INTEREST. ^ 2D METHOD. 255. There is another rule resulting from the last analysis, which is regarded as the best general method of computing interest. RULE. I. Find the interest for 1 year and divide it bylZ: the quotient will be the interest for 1 month. II. Multiply the interest for 1 month by the time expressed in months and parts of a month, and the product will be the required interest. NOTE, Since a month is reckoned at 30 days, any number of days is reduced to decimals of a month by dividing the days by 3. EXAMPLES. 1. What is the interest of $327.50 for 3 years 7 months and 13 days, at 7 per cent ? OPERATION. 3yrs.=3Qmos. $327.50 7mos. .07 13 days A\mos. 12)22.9250 =int. for 1 year. Timer=43.4jwos. 1.9104 + =int. for 1 month. NOTE. The method em- 43.4^ =time in months, ployed, and the number of 6368 decimal places used, in com- puting interest, may affect the mills, and possibly, the last figure in cents. It is best 7 64 1 6 to use 4 places of decimals. $32.97504 Ans. 2. What is the interest of $1728.60, at 7 per cent, for 2 years 6 months and 21 days ? 3. What is the interest of $288.30, at 7 per cent, for I year 8 months and 27 days ? 4. What is the interest of $576.60, at 6 per cent, for 10 months aucl 18 days? 5. What is the interest of $854.42, at 6 per cent, for 3 months and 9 days ? 6. What is the interest of $1153.20, at 6 per cent, for I 1 months and 6 days ? 255. How do you find the interest for years, months and days by the second method ? SIMPLE INTEREST. 245 7. What is the interest of $2306.54, at 5 per ceut, for 7 months and 28 days ? 8. What is the interest of $4272.10, at 5 per cent, for 10 months and 28 days? 9. What is the interest of $1620, at 4 per cent, for 5 years and 24 days ? 10. What is the interest of $2430.72, at 4 per cent, for 10 years and 4 months ? 11. What is the interest of $3689.45, at 7 per cent, for 4 years and 7 months ? 12. What is the interest 01 $2945.96, at 7 per cent, for 7 years and 3 days ? 13. W T hat is the interest, at 8 per cent, of $675.89, for 3 years 6 months and 6 days ? 14. What is the interest, at 8 per cent, on $12324, for 3 years and 4 months ? 15. What is the interest, at 9 per cent, on $15328.20, for 4 years and 7 months ? 16. What is the interest of $69450 for 1 year 2 months and 12 days, at 9 per cent ? 17. What is the interest of $216.984 for 3 years 5 months and 15 days, at 10 per cent ? 18. What is the interest of $648.54 for 7 years 6 months, at 4J per cent ? 19. What is the interest of $1297.10 for 8 years 5 months, at 5 1 per cent ? 20. What is the interest of $864.768 for 9 months 25 days, at 6 \ per cent ? 21. What is the interest of $2594.20 for 10 months and 9 days, at 7 1 per cent? 22. What is the amount of $2376.84 for 3 years 9 months and 12 days, at 8 J per ceut ? 23. What is the amount of $5148.40 for 7 years 11 months and 23 days, at 9 J per cent ? 24. What is the amount of $3565.20 for 3 years 9 months, at 10 J per cent? 24:6 SIMPLE INTEREST. 25. What is the amount of $125.75 for 1 year 9 months and 27 days, at 7 per cent ? 26. What is the amount of $256 for 10 months 15 days, at 7 J per cent ? 27. What is the interest on a note of $264.42, given Janu- ary 1st, 1852, and due Oct. 10th, 1855, at 4 per cent? 28. Gave a note of $793.26 April 6th, 1850, on interest at 7 per cent : what is due September 10th, 1852 ? 29. What amount is due on a note of hand given June 7th, 1850, for $512.50, at 6 per cent, to be paid Jan. 1st, 1851 ? 30. What is the interest on $1250.75 for 90 days, at 10 per cent ? 31. What is the amount of $71.09 from Feb. 8th, 1848, to Dec. 7th, 1852, at 6 j per cent ? 32. What will be due on a note of $213.27 on interest after 90 days, at 7 per cent, given May 19th, 1836, and pay- able October 16th, 1838 ? 33. What is the interest of $426.54, from August 15th, 1837, to March 13th, 1840, at 7 per cent? 34. What is the interest of $2132.70, from Nov. 17th, 1838, to Feb. 2d, 1839, at 7J per cent? 35. What is the interest of $38463, from April 27th, 1815, to Sept. 2d, 1824, at 8 per cent ? 36. What is the interest of $14231.50, from June 29th, 1840, to April 30th, 1845, at 8J per cent? 37. What is the interest of $426.50, from Sept. 4th, 1843, to May 4, 1849, at 9 per cent? 38. What is the interest of $4320, from Dec. 1st, 1817, to Jan. 22d, 1833, at 9J per cent?" 39. What is the amount of $397.16, from March 24, 1824, to March 31st, 1835, at 10| per cent ? 40. What is the amount of $328.12, from July 4th, 1809, to Feb. 15th, 1815, at 3 per cent ? 41. What is the amount of $164.60, from Sept. 27th, 1845, to March 24, 1855, at 1J per cent? 42. What is the amount of $1627.50, from July 4th, 1839, to August 1st, 1855, at 8 per cent? PARTIAL PAYMENTS. 24:7 CASE III. 256. When the principal is in pounds, shillings and pence. 1. What is the interest, at 7 per cent, of 27 15s. 9d., for 2 years ? OPERATION. ANALYSIS. The interest on pounds 27 15s. 9J. = 27.7875 and decimals of a pound is found in Q>J the same way as the interest on dol- lars and decimals of a dollar: after 1.945125 which the decimal part of the interest 2 may be reduced to shillings and Ans. 3 178. 1. Reduce the shillings and pence to the decimal of a pound and annex the result to the pounds. II. Find the interest as though the sum were United States Money, after which reduce the decimal part to shil- lings and pence. 2. What is the interest of 67 19s. Qd. } at 6 per cent, for 3 years 8 months 16 days ? 3. What is the interest of 127 15s. 4d., at 6 per cent, for 3 years and 3 months ? 4. What is the interest of 107 16s. IQd., at 7 per cent, for 3 years 6 months and 6 days ? 5. What will 279 13s. 8d. amount to in 3 years and a half, at 5J per cent per annum? PARTIAL PAYMENTS. 257.. A PARTIAL PAYMENT is a payment of a part of a note or bond. We shall give the rule established in New York (see Johnson's Chancery Reports, vol. I. page 17), for computing the interest on a bond or note, when partial payments have been made. The same rule is also adopted in Massachusetts, and in most of the other states. 256. How do you find the interest when the principal is in pounds, shillings and pence ? 248 PARTIAL PAYMENTS. RULE. I. Compute the interest on the principal to the time of the first payment, and if the payment exceed this interest, add the interest to the principal and from the sum subtract the payment : the remainder forms a new principal : II. But if the payment is less than the interest, take no notice of it until other payments are made, which in all, shall exceed the interest computed to the time of the last payment : then add the interest, so computed, to the princi- pal, and from the sum subtract the sum of the payments : the remainder will form a new principal on which interest is to be computed as before. NOTE In computing interest on notes, observe that the day on which a note is dated and the day on which it falls due, are not both reckoned in determining the time, but one of them is always excluded. Thus, a note dated on the first day of May and falling due on the 16th of June, will bear interest but one month and 1 5 days. EXAMPLES. $349.998 Buffalo, May 1st, 1826. 1. For value received, I promise to pay James Wilson or order, three hundred and forty-nine dollars ninety-nine cents and eights mills with interest at 6 per cent. James Pay well. On this note were endorsed the following payments : Dec. 25th, 1826 Received $49.998 July 10th, 1827 " $ 4.998 Sept. 1st, 1828 " $15.008 June 14th, 1829 " $99.999 What was due April 15th, 1830 ? Principal on int. from May 1st, 1826, - - - - $349.998 Interest to Dec. 25th, 1826, time of first pay- ment, 7 months 24 days 13.649 + Amount, - - - $363.647 257. What is a partial payment? What is the rule for computing Interests when there are partial payments ? PARTIAL PAYMENTS. 249 Payment Dec. 25th, exceeding interest then due $ 49.998 Remainder for a new principal $313.649 Interest of $313.649 from Dec. 25, 1826, to June 14th, 1829, 2 years 5 months 19 days, - $ 46.4721 Amount "$360.1211 Payment, July 10th, 1827, .less than {* ^ QQO interest then due ) * ' Payment, Sept. 1st, 1828 15.008 Their sum less than interest then due - $20.006 Payment, June 14th, 1829 - - - - 99.999 Their sum exceeds the interest then due- - - $120.005 Remainder for a new principal, June 14, 1829, $240.1161 Interest of $240.168 from June 14th, 1829, to April 15th, 1830, 10 months 1 day - - - $ 12.0458 Total due, April 15th, 1830 - -"$252.1619 + $3469.327 New York, Feb, 6, 1825. 2. For value received, I promise to pay William Jenks, or order, three thousand four hundred and sixty-nine dollars and thirty-two cents, with interest from date, at 6 per cent. Bill Spendthrift. On this note were endorsed the following payments : May 16th, 1828, received $ 545.76 May 16th, 1830, " $1276.00 Feb. 1st, 1831, " $2074.72 What remained due Aug llth, 1832 ? 3. A's note of $635.84 was dated September 5, 1817, on which were endorsed the following payments, viz. : Nov. 13th, 1819, $416.08 ; May 10th, 1820, $152.00 : what was due March 1st, 1821, the interest being 6 per cent? LEGAL INTEREST, 258. Legal Interest is the interest which the law permits a person to receive for money which he loans, and the laws do not favor the taking of a higher rate. In most of the States the rate is fixed at 6 per cent ; in New York, South Carolina and Georgia, it is 7 ; and in some of the States the rate is fixed as high as 10 per cent 250 PROBLEMS IN INTEREST. PROBLEMS IN INTEREST. 259. In all questions of Interest there are four things con- sidered, viz. : 1st, The principal ; 2d, The rate of interest ; 3d, The time ; and &th, The amount of interest. If three of these are known, the fourth can be found, I. Knowing, the principal, rate, and time, to find the inter- est. This case has already been considered. II. Knowing the interest, time, and rate, to find the prin- cipal. Cast the interest on one dollar for the given time, and then divide the given interest by it the quotient ivill be the princi- pal. III. Knowing the interest, the principal, and the time, to find the rate. Cast the interest on the principal for the given time at 1 per cent and then divide the given interest by it the quotient will be the rate of interest. IV Knowing the principal, the interest, arid the rate, to find the time. Cast the interest on the given principal at the given rate for 1 year and then divide the interest by it the quotient will be the time in years and decimals of a year. EXAMPLES 1. The interest of a certain sum for 4 years, at 7 per cent, is $266 : what is the principal? 2. The interest of $3675, for 3 years, is $171.15 : what is the rate? 3. The principal is $459, the interest $183.60, and the rate 8 per cent : what is the time ? 4. The interest of a certain sum, for 3 years, at 6 per cent, is $40.50 : what is the principal ? 5. The principal is $918, the interest $269.28, and the rate 4 per cent : what is the time ? 258. What is legal interest ? 259. How many things are considered in every question of interest? What arc they ? What is the rule for each ? COMPOUND INTEKEST. 251 COMPOUND INTEREST. 260. Compound Interest is when the interest on a princi- pal, computed to a given time, is added to the principal, and the interest then computed on this amount, as on a new principal. Hence, Compute the interest to the time at which it becomes due ; then add it to the principal and compute the interest on the amount as on a new principal: add the interest again to the principal and compute the interest as before ; do the same for all the times at which payments of interest become due ; from the last result subtract the principal, and the remainder will be the compound interest. EXAMPLES. 1. What will be the compound interest, at 7 per cent, of $3150 for 2 years, the interest being added yearly? * OPERATION. $3750.000 principal for 1st year. $3750 x. 07= 262.500 interest for 1st year 4012.500 principal for 2d " $4012.50 x. 07= 280.875 interest for 2d " 4293.375 amount at 2 years. 1st principal 3750.000 Amount of interest $543.375. 2. If the interest be computed annually, what will be the compound interest on $100 for 3 years, at 6 per cent? 3. What will be the compound interest on $295.37, at 6 per cent, for 2 years, the interest being added annually ? 4. What will be the compound interest, at 5 per cent, of $1875, for 4 years? 5. What is the amount at compound interest of $250, for 2 years, at 8 per cent ? 6. What is the compound interest of $939.64, for 3 years, at 7 per cent ? 7. What will $125.50 amount to in 10 years, at 4 per cent compound interest ? 260. What of compound interest ? How do you compute it ? 252 COMPOUND INTEREST. NOTE. The operation is rendered much shorter and easier, by taking the amount of 1 dollar for any time and rate given in the following table, and multiplying it by the given principal ; the product will be the required amount, from which subtract the given principal, and the result will be the compound interest.* TABLE. Which shows the amount of $1 or 1, compound interest, from 1 year to 20, aud at the rate of 3, 4, 5, 6, and 7 per cent. Years. jiSper cent. 4 per rent.io per cent. ti per cent. ' per ci-Bt. Vars. 1 1.03000 1.04000 1.05000 1.06000 1.07000 1 2 1.0(5090 1.08160 1.10250 1.12360 1.14490 2 3 1.09272 1.12486 1.15762 1.19101 1.22504 3 1.135501.109851.21550 1.26247 1.31079 4 5 1.15927 1.216'>5 1.27628 1. 33822 1.40255 5 6 1 19405 1.26531 1.34009 1.41851 1.50073 6 7 1.22987 1.31593 1. 40710 1.50363 1.60578 7 8 ft.26677 9 1.30477 1.36856! 1.47745 1.4233111.55132 1.59384 1.C8947 1.71818 1.83845 8 9 10 1.34391 1.480:38 1.62889 1.79084 1.96715 10 11 11.38433 1.53945 1.71033 1.89829 2.10485 11 12 1.4257(5 1.60103 1.79585 2.012192.25219 12 13 1.46853 1.66507 1.88564 2.13292240984 13 14 1.5! 258 1.73167 1.97993 2.260902,57853 14 15 1.55796 1.80094 2.07892 2.396552.75903 15 16 1.60470 1.8729812.18287 2.54035 2.95216 16 17 1. (55284 1.94790J2. 29201 2. 69277 ; 3. 15881 17 18 1.70243 2.02581 2.40661 2.854333.37993 18 19 1.75350 2.10684 2.52695 3.025593.61652 19 20 1.80611 2.19112 ,2. (55329 3. 2071 3 i 3. 86968 1 20 NOTE. When there are months and days in the time, find the amount for the years, and on this amount cast the interest for the mcnths and days : this, added to the last amount, will be the re- quired amount for the whole time. 8. What is the amount of $96.50 for 8 years and 6 months, interest being compounded annually at 7 per cent ? 9. What is the compound interest of $300 for 5 years 8 months and 15 days, at 6 per cent ? 10. What is the compound interest of $1250 for 3 years 3 months and 24 days, at 7 per cent ? 11. What will $56.50 amount to in 20 years and 4 months, at 5 per .cent compound interest ? * The result may differ in the mills place from that obtained by the other rule. DISCOUNT. 253 DISCOUNT. x 261. DISCOUNT is an allowance made for the payment of money before it is due. THE FACE of a note is the amount named in the note.* NOTE. DAYS OP GRACE are days allowed for the payment of a note after the expiration of the time named on its face. By mercantile usage a note does not legally fall due until 3 days after the expiration of the time named on its face, unless the note specifies without grace. Days of grace, however, are generally confined to mercantile paper and to notes discounted at banks. 262. The PRESENT VALUE of a note is such a sum as being put at interest until the note becomes due, would increase to an amount equal to the face of the note. The discount on a note is the difference between the face of the note and its present value. 1. I give my note to Mr. Wilson for $10 7, payable in 1 year : what is the present value of the note if the interest is 7 per cent. ? what the discount ? OPERATION. ANALYSIS. Since 1 dollar in 1 year $107 -f- 1,07 $100. at 7 per cent, will amount to $1.07, the PROOF present value will be as many dollars y n 4. (frinn 1,1. <6 *r as $1.07 is contained times in the face t, . \, ^ \ n A of the note: viz., $100: and the dis- -Principal, count will be $107- $100= $7: hence, Amount, $107 Discount, 7 Divide the face of the note by 1 dollar plus the interest of 1 dollar for the given time, and the quotient will be the pre- sent value : take this sum from the face of the note and the remainder will be the discount. 261. What is discount ? What is the face of a note ? What are days of grace? 362. What is present value ? What is the discount ? How do you find the present value of a note ? * See Appendix, page 3l(X 254 DISCOUNT. EXAMPLES. 1. What is the present value of a note for $1828,75, eke in 1 year, and bearing an interest of 4 J per cent ? 2. A note of $1651.50 is due in 11 months, but the person to whom it is payable sells it with the discount off at 6 per cent : how much shall he receive ? NOTE. When payments are to be made at different times, find the present value of the sums separately, and tfieir sum will be the present value of the note. 3 What is the present value of a note for $10500, on which $900 are to be paid in 6 mouths ; $2700 in one year ; $3900 in eighteen months ; and the residue at the expiration of two years, the rate of interest being 6 per cent per annum ? 4. What is the discount of <4500, one-half payable in six months and the other half at the expiration of a year, at 7 per cent per annum ? 5. What is the present value of $5760, one-half payable in 3 months, one-third in 6 months, and the rest in 9 months, at 6 per cent per annum ? 6. Mr. A gives his note to B for $720, one-half payable in 4 months and the other half in 8 months ; what is the present value of said note, discount at 5 per cent per annum ? 7. What is the difference between the interest and discount of $750, due nine months hence, at 7 per cent ? 8. What is the present value of $4000 payable in 9 months, discount 4J per cent per ami am ? 9. Mr. Johnson has a note against Mr. Williams for $2146.50, dated August 17th, 1838, which becomes due Jan. llth, 1839 : if the note is discounted at 6 per cent, what ready money must be paid for it September 25th, 1838 ? 10. C owes D $3456, to be paid October 27th, 1842 ; C wishes to pay on the 24th of August, 1838, to which D con- sents ; how much ought D to receive, interest at 6 per cent ? 11. What is the present value of a note of $4800, due 4 years hence, the interest being computed at 5 per cent per annum ? 12. A man having a horse for sale, offered it for $225 cash in hand, or $230 at 9 months ; the buyer chose the latter : did the seller lose or make by his offer, supposing money to be worth 7 per cent ? BANK DISCOUNT, 255 BANK DISCOUNT. 263. BANK DISCOUNT is the charge made by a bank for the payment of money on a note before it becomes due. By the custom of banks, this discount is the interest on the amount named in a note, calculated from the time the note is discounted to the time when it falls due ; in which time the three days of grace are always included. The interest is always paid in advance. RULE Add 3 days to the time which the note has to run, and then calculate the interest for that time at the given rate. EXAMPLES. 1. What is the Dank discount of a note for $350, payable 3 months after date, at 7 per cent interest ? 2. What is the bank discount of a note of $1000 payable in 60 days, at 6 per cent interest ? 3. A merchant sold a cargo of cotton for $15720, for which he receives a note at 6 months : how much money will he receive at a bank for this note, discounting it at 6 per cent interest ? 4. What is the bank discount on a note of $556. 2 1 paya- ble in 60 days, discounted at 6 per cent interest? 5. A has a note against B for $3456, payable in three months ; he gets it discounted at 7 per cent interest : how much does he receive ? 6. What is the bank discount on a note of 367.47, having 1 year, 1 month, and 13 days to run, as shown by the face of the note, discounted at 7 per cent ? 7- For value received, I promise to pay to John Jones, on the 20th of November next, six thousand five hundred and seventy-nine dollars and 15 cents. What will be the discount on this, if discounted on the 1st of August, at 6 per cent per annum ? 263. What is bank discount ? How is interest calculated by the custom of banks ? How is the interest paid ? How do you find the interest ? 256 BANK DISCOUNT. 8. A merchant bought 115 barrels of flour at $7.50 cents a barrel, and sells it immediately for $9.75 a barrel, for which he receives a good note, payable in 6 months. If he should get this note discounted at a bank, at 6 per cent, what will be his gain on the flour ? 264. To make a note due at a future lime, whose present value shall be a given amount. 1. For what sum must a note be drawn at 3 months, so that when discounted at a bank, at 6 per cent, the amount received shall be $500 ? ANALYSIS If we find the interest on 1 dollar for the given time, and then subtract that interest from 1 dollar, the remainder will be the present value of 1 dollar, due at the expiration of that time. Then, the number of times which the present value of the note contains the present value of 1 dollar, will be the num- ber of dollars for which the note must be drawn : hence, Divide the present value of the note by the present value of 1 dollar, reckoned for the same time and at the same rate of interest , and the quotient will be the face of the note, OPERATION. Interest of $1 for the time, 3mo. and Ma. =$0.0155, which taken from $1, gives present value of $1=0.9845; then, $500^- 0.9845= $507.872-1- =face of note. PROOF. Bank interest on $507.872 for 3 months, including 3 days of grace, at 6 per cent =7.872, which being taken from the face of the note, leaves $500 for its present value, EXAMPLES, 1 . For what sum must a note be drawn, at 7 per cent, payable on its face in 1 year 6 months and 1 5 days, so that when discounted at bank it shall produce $307.27 ? 2. A note is to be drawn having on its face 8 months and 1 2 days to run, and to bear an interest of 7 per cent, so that it will pay a debt of $5450 : what is the amount ? 364. How do you make a note payable at a future time, whose pre- sent value shall be a given amount ? EQUATION OF PAYMENTS. 257 3. What sum, 6 months and 9 days from July 18th, 1856, drawing an interest of 6 per cent, will pay a debt of $674.89 at bank, on the 1st of August, 1856 ? 4. Mr Johnson has Mr. Squires' note for $814.57, having 4 months to run, from July 13th, without interest. On the first of October he wishes to pay a debt at bank of $750.25, and discounts the note at 5 'per cent in payment : how much must he receive back from the bank ? 5. Mr. Jones, on the 1st of June, desires to pay a debt at bank by a note dated May 1 6th, having 6 months to run and drawing 7 per cent interest : for what amount must the note be drawn, the debt being $1683.75 ? 6 Mr. Wilson is indebted at the bank in the sum of $367.464, which he wishes to pay by a note at 4 months with interest at 7 per cent : for what amount must the note be drawn ? EQUATION OF PAYMENTS, 265. EQUATION OF PAYMENTS is the operation of finding the mean time of payment of several sums due at different times, so that no interest shall be lost or gained.* 1. If I owe Mr. Wilson 2 dollars to be paid in 6 months, 3 dollars to be paid in 8 months, and 1 dollar to be paid in 12 months, what is the mean time of payment ? OPERATION. Int. of $2 for 6rao.=int. of $1 for 12mo. 2x 612 " of $3 for 8rao; int. of $1 for 24??io. 3x 8 = 24 " of $1 for 12wio.=mt. of $1 for 12 mo. I x 12^12 $6 48 48 ANALYSIS. The interest on all the sums, to the times of pay- ment, is equal to the interest of $1 for 48 months. But 48 is equal to the sum of all the products which arise from multiplying each sum by the time at which it becomes duo: hence, the sum of the products is equal to the time which would be necessary for $1 to produce the game interest as would be produced by all the principals. * The mean time of payment is sometimes found by first finding the jyrcsent value of each payment ; but the rule here given has the sanc- tion of the best authorities in this country and England. 17 253 EQUATION OF PAYMENTS. ' $1 will produce a certain interest in 48 months, in what time will $6 (or the sum of the payments) produce the same interest ? The time is obviously found by dividing 48 (the sum of the pro- ducts) by $6, (the sum of the payments.) Hence, to find the mean time, Multiply each payment by the time before it becomes due, and divide the sum of the products by the sum of the pay- ments : the quotient will be the mean time. EXAMPLES. 1. B. owes A $600 ; $200 is to be paid in two months, $200 in four months, and $200 in six months : what is the mean time for the payment of the whole ? OPERATION. 200x2-= 400 ANALYSIS. We here multiply each 200x4 800 sum by the time at which it becomes QHA f_ionn due, and divide the sum of the products JUU by the sum of the payments. 6|00 )24|00 Ans. 4 months. 2. A merchant owes $600, of which $100 is to be paid in 4 months, $200 in 10 months, and the remainder in 16 months : if he pays the whole at once, in what time must he make the payment ? 3. A merchant owes $600 to be paid in 12 months, $800 to be paid in 6 months, and $900 to be paid in 9 months : what is the equated time of payment ? 4. A owes B $600 ; one-third is to be paid in 6 months, one-fourth in 8 months, and the remainder in 12 months : what is the .mean time of payment ? 5. A merchant has due him $300 to be paid in 60 days, $500 to be paid in 120 days, and $750 to be paid in 180 days : what is the equated time for the payment of the 6. A merchant has due him $1500 : one-sixth is to bo paid in 2 months, one-third in 3 months, and the rest in 6 months : what is the equated time for the payment of the whole ? 265. What is equation of payments ? How do you find the mean or equated time ? EQUATION OF PAYMENTS. 259 7. I owe $1000 to be paid on the first 'of January, $1500 on the 1st of February, $3000 on the 1st of March, and $4000 on the 15th of April : reckoning from the 1st of Janu- ary, and calling February 28 days, on what day must the money be paid ? NOTE. If one of the payments, as in the above example, is due on the day from which the equated time is reckoned, its corres- ponding product will be notliing, but the payment must still be added in finding the sum of the payments, 8. I owe Mr Wilson $100 to be paid on the 15th of July, $200 on the 15th of August, and 300 on the 9th of Septem- ber : what is the mean time of payment ? OPERATION From 1st of July to 1st payment 14 days " " " to 2d payment 45 days. " to 3d payment 70 days. 100x14= 1400 200x45= 9000 Tlien by rule given above we 300 X 70 = 2 1 000 have, 600 6|00)314|00 fili Hence, the equated time is 52^ days from the 1st of July ; that is, on the 22d day of August. But if we estimate the time from the 15th of July we shall have From July 15th to 1st payment days. " " to 2d payment 30 days. " " to 3d payment 54 days. Then, 100 x 0= 000 200x30= COOO 300x54 = 16200 600 Hence, the payment is due in 37 days from July 15th; or, on the 22d of August the same as before. Therefore : Any day may be taken as the one from, which the mean time is reckoned. NOTE. If one payment is due on the day from which the time is reckoned, how do you treat it ? Can you compute the time from any day? 260 ASSESSING TAXES. 9. Mr. Jones purchased of Mr. Wilson, on a credit of six months, goods to the following amounts : 15th of January, a bill of $3t50, 10th of February, a bill of 3000, 6th of March, a bill of 2400, 8th of June, a bill of 2250. He wishes, on the 1st of July, to give his note for the amount : at what time must it be made payable ? 10. Mr Gilbert bought $4000 worth of goods ; he was to pay $1600 in five months, $1200 in six months, and the re- mainder in eight months : what will be the time of credit, if he pays the whole amount at a single payment ? 11. A merchant bought several lots of goods, as follows : A bill of $650, June 6th, A bill of 890, July 8th, A bill of 7940, August 1st. Now, if the credit is 6 months, how many days from De- cember 6th before the note becomes due ? At what time ? ASSESSING TAXES. 26G. A tax is a certain sum required to be paid by the inhabitants of a town, county, or state, for the support of government or some public object. It is generally collected from each individual, in proportion to the amount of his property. In some states, however, every white male citizen over the age of twenty-one years is required to pay a certain tax. This tax is called a poll-tax ; and each person so taxed is called a poll. 267. In assessing taxes, the first thing to be done is to make a complete inventory of all the property in the town on which the tax is to be laid. If there is a poll-tax, make a full list of the polls and multiply the number by the tax on each poll, and subtract the product from the whole tax to be 266. What is a tax ? llow is it generally collected ? What is a poll-tax ? ASSESSING TAXES. 2C1 raised by the town : the remainder will be the amount to be raised on the property Having done this, divide the whole tax to be raised by the amount of taxable properly and the quotient will be the tax on $1. Then multiply this quotient by the inventory of each individual, and the product will be the tax on his property EXAMPLES. 1. A certain town is to be taxed $4280 ; the property on which the tax is to be levied is valued at $1000000. Now there are 200 polls, each taxed $1.40. The property of A is valued at $2800, and he pays 4 polls. B's at $2400, pays 4 polls. E's at $7242, pays 4 polls. C's at $2530, pays 2 " F's at $1651, pays 6 " D's at $2250, pays 6 " G's at $1600.80 pays 4 " What will be the tax on 1 dollar, and what will be A's tax, and also that of each on the list ? First; $1.40 x 200 = $280 amount of poll-tax. $4280 $280 4000 amount to be levied on property. Then, $4000-i-$1000000=4 mills on $1. Now, to find the tax of each, as A's, for example, A's inventory $2800 _^004 TT200 4 polls at $1,40 each - - 5.60 A's whole tax - - - - - $16.800> In the same manner the tax of each person in the town- ship may be found. Having found the per cent, or the amount to be raised on each dollar, form a table showing the amount which certain sums would produce at the same rate per cent. Thus, after having found, as in the last example, that 4 mills are to be raised on every dollar, we can, by multiplying in succession by the numbers 1, 2, 3, 4. 5, 6, 7, 8, &c., form the following 267. What is the first thing to be done in assessing a tax ? If there is a poll-tax, how do you find the amount ? How x then do you find the per cent of tax to be levied on a dollar ? How do you then find the amount to be levied on each individual ? 262 ASSESSING TAXES. TABLE $ $ $ $ $ $ 1 gives 0.004 20 gives 080 300 gives 1.200 2 " 0.008 30 " OJ20 400 " 1.600 3 " 0-012 40 (< 0.160 500 " 2.000 4 " 0.016 50 " 0.200 600 " 2.400 5 " 0.020 60 " 0.240 700 " 2.800 6 " 0.024 70 " 0.280 800 " 3.200 7 " 0.028 80 " 0.320 900 " 3.600 8 " 0.032 90 " 0.360 1000 " 4.000 9 " 0.036 100 " 0.400 2000 " 8.000 10 " 0.040 200 " 0.800 3000 " 12.000 This table shows the amount to be raised on each sum in the columns under $'s. EXAMPLES. 1. Find the amount of B's tax from this table. B's tax on $2000 - - is - $8.000 B's tax on 400 - - is, - $1.600 B's tax on 4 polls, at $1.40 - $5 600 B's total tax - is - $15.200 2. Find the amount of C's tax from the table. C's tax on $2000 - - is - $8.000 C's tax on 500 - - is - $2.000 C's tax on 30 - - is - $0.120 C's tax on 2 polls - - is - $2.800 C's total tax - - is -"$12.920 In a similar manner, we might find the taxes to be paid by D, E, &c. 3. If the people of a town vote to tax themselves $1500, to build a public hall, and the property of the town is valued at $300.000, what is D's tax, whose property is valued at $2450? 4. In a school district a school is supported by a tax on the property of the district valued at $121340. A teacher is employed for 5 months at $40 a month, and contingent ex- penses are $42,68 ; what will be a farmer's tax whose property is valued at $3125? COINS AND CURRENCY. 263 COINS AND CURRENCY. 268. Coins are pieces of metal, of gold, silver, or copper, of fixed values, and impressed with a public stamp prescribed by the country where they are made. These are called specie, and are declared to be a legal tender in payment of debts. 2(51). Currency is what passes for money. In our country there are four kinds. 1st. The coins of the country : M. Foreign coins, having "a fixed value established by law : 3e?. Bank notes, redeemable in specie. 4th. Paper money declared a legal tender, by act of Congress. NOTE. The foreign coins most in use in this country are the English shilling, valued at 22 cents 2 mills ; the English sove- reign, valued at $4,84 ; the French franc, valued at 18 cents 6 mills ; and* the five-franc piece, valued at $0.93. Although the currency of the United States is in dollars, cents and mills, yet in some of the States accounts are still kept in pounds, shillings and pence. In all the States the shilling is reckoned at 12 pence, the pound at 20 shillings, and the dollar at 100 cents. The following table shows the number of shillings in a dol- lar, the value of 1 in dollars, and the value of $1 in the fraction of a pound ? In English currency, 4s. bd. - 1=$4.84 : , and$l= T .-i- T . In N. E., Ya , Ky., ( C ^1 &31 , , Tenn., j $ 5, an * ^TtT- In N. Y., Ohio, N. [ Carolina-, j 8s. - l=$2 2 , and $1 |. In N. J., Pa., Del., [ Md., ) Ts. &d. - J61=$2|, and$l= f In S. Carolina &Ga. 4s. Sd. - l:=$4f, and $!=.,&. In Canada & Nova ) Scotia, j 5., - 1=*, and $!=: l. 368. What arc coins? V/hat arc they called ? Wliat is made " legal tender? 26 tt REDUCTION OF CURRENCIES. REDUCTION OF CURRENCIES. 270. Reduction of Currencies is changing their denomina- tions without changing their values. There are two cases of the Reduction of Currencies : 1st. To change a currency in pounds, shillings and pence, to United States currency. 2d. To change United States currency to pounds, shillings and pence. 271. To reduce pounds, shillings and pence to United States currency. 1. What is the value of <3 12s. Qd., New England cur- rency, in United States money. OPERATION. ANALYSIS. Since l = $3i the 3 12s. Qd.=3.G%5 number of dollars in 3 12s. Gd.= rlnllc in ^1 - 31 3.625, will be equal to 3.625 taken 3^ times : that is, to $12,08 : 1.2084" hence, 10.875 Ans. $12.083 + Multiply the amount reduced to pounds and the decimals of a pound by the number of dollars in a pound, and the product will be the answer. 272. To reduce United States money to pounds, shillings and pence. 1. What is the value of $375.81, in pounds, shillings and pence, New York currency ? ANALYSIS. Since $!=, the number of pounds in $375.87 will bo OPERATION. equal to this number taken times : $375.87 X -? =<150 348 that is, equal to 150.348=150 6s. =^E150 6s Hid . : hence, 200. What is currency ? How many kinds arc there ? What foreign coins are most used in this country? What are the denominations of United States currency ? What denominations are sometimes used in the States ? 270. What is reduction of currencies ? How many kinds of reduc- tion arc there ? What arc they ? 271. What is the rule for reducing from pounds, shillings rind pence to United States money ? EXCHANGE. 265 Multiply the amount by that fraction of a pound which denotes the value q/ 1 $1, and the product will be the answer in pounds and decimals of a pound. EXAMPLES 1. What is the value of 127 18s. 6d., New England currency, in United States money ? 2. What is the value of $2863.75 in pounds, shillings and pence, Pennsylvania currency ? 3. What is the value of 459 3s. Qd., Georgia currency, in United States money ? 4. What is the value of $973.28 in pounds, shillings and pence, North Carolina currency ? 5. What is the value in United States money of 637 18s. 8d., Canada currency ? 6. Reduce $102.85 to English money ; to Canada cur- rency ; to New England currency ; to New York currency ; to Pennsylvania currency ; to South Carolina currency. 7. Reduce 51 13s. OJtf. English money ; 62 10s. Can- ada currency ; 75 New England currency ; 100 New York currency ; 193 15s. Pennsylvania currency ; and 58 6s. 7Jrf. Georgia currency, to United States money. EXCHANGE. 273. EXCHANGE denotes the payment of a sum of money by a person residing in one place to a person residing in an- other. The payment is usually made by means of a bill of exchange. A BILL OF EXCHANGE is an order from one person to another directing the payment to a third person named therein of a certain sum of money : 1. He who writes the open letter of request is called the drawer or maker of the bill. 2. The person to whom it is directed is called the draw'ee. 272. What is the rule for reducing from United States money to pounds, shillings and pence ? 273. What does exchange denote ? How is the payment generally made ? What is a bill of exchange ? Who is the drawer ? Who the drawee ? Who the buyer or remitter ? 266 FOREIGN BILLS. 3. The person to whom the money is ordered to be paid is called the payee ; and 4. Any person who purchases a bill of exchange is called the buyer or remitter. 274. A bill of exchange is called an inland bill, when the drawer and drawee both reside in the same country ; and when they reside in different countries, it is called a foreign bill. Exchange is said to be at par, when an amount at the place from which it is remitted will pay an equal amount at the place to which it is remitted. Exchange is said to be at a premium, or above par, when the sum to be remitted will pay less at the place to which it is remitted ; and at a dis- count, or below par, when it will pay more. EXAMPLES. 1. A merchant at Chicago wishes to pay a bill in New York amounting to $3675, and finds that exchange is 1J per cent premium : what must he pay for his bill? 2. A merchant in Philadelphia wishes to remit to Charles- ton $8756.50, and finds exchange to be 1 per cent below par ; what must he pay for the bill ? 3. A merchant in Mobile wishes to pay in New York $6584, and exchange is 2| per cent premium : how much must he pay for such a bill '{ 4. A merchant in Boston wishes to pay in New Orleans $4653.75 ; exchange between Boston and New Orleans is 1J per cent below par : what must he pay for a bill ? 5. A merchant in New York has $3690 which he wishes to remit to Cincinnati ; the exchange is 1 \ per cent below par : what will be the amount of his bill ? FOREIGN BILLS. 275. A Foreign Bill of Exchange is one in which the drawer and drawee live in different countries. NOTE. In all Bills of Exchange on England, the sterling is the unit or base, and is still reckoned at its former value of $4$ = $4.4444 -f, instead of its present value $4.84. 274. When is a bill of exchange said to be inland ? When foreign ? When is exchange said to be at par ? When at a premium ? When at a discount ? FOREIGN BILLS. 267 Hence, 1 =$4.4444 -f Add 9 per cent, .3999 Gives the present value of 1 $4.8443. Hence, the true par value of Exchange on England is 9 per cent on the nominal base. 1. A merchant in New York wishes to remit to England a' bill of Exchange for 125 15s. Qd : how much must he pay for this bill when exchange is at 9J per cent premium? 125 15s. 6d. ...... =125.775 Add 9| per cent ..... gives amount in 's, at $4f== NOTE. The pounds and decimals of a pound are reduced to dollars by multiplying by 40 and dividing by 9 giving, in this case, $612.105. RULE. I. Reduce the amount of the bill to pounds and decimals of a pound, and then add the premium of exchange. II. Multiply the result by 40 and divide the product by 9 : the quotient will be the answer in United States Money. 2. A merchant shipped 100 bales of cotton to Liverpool, each weighing 450 pounds. They were sold at *l\d. per pound, and the freight and charges amounted to 187 10s. He sold his bill of exchange at 9} per cent premium : how much should he receive in United States Money ? 3. There were shipped from Norfolk, Ya., to Liverpool, Sbhhd. of tobacco, each weighing 450 pounds. It was sold at Liverpool for l^^d. per pound, and the expenses of freight and commissions were 92 Is. Sd. If exchange in New York is at a premium of 9J per cent, what should the owner receive for the bill of exchange, in United States Money ? 276. The unit or base of the French Currency is the French franc, of the value of 18 cents 6 mills. The franc is divided into tenths, called decimes, corresponding to our dimes, and into centimes corresponding to cents. Thus, 5.12 is read, 5 francs and 12 centimes. 275. What is a foreign bill of exchange ? In bills on England, wh.it is the unit, or base? What is the exchange value of the sterling ? How much is the true value above the commercial value of the ster- ling? How do you find the value of a bill in English currency in United States mo'ney? 268 DUTIES. All bills of exchange on France are drawn in francs. Exchange is quoted in New York at so many francs and centimes to the dollar. 1. What will be the value of a bill of exchange for 4^36 francs, at 5,25 to the dollar ? ANALYSIS. Since 1 dollar will buy 5.25 francs, the bill will cost as many OPERATION. dollars as 5.25 is contained timesin the 5.25)4536($864 Ans amount of the bill ; hence, Divide the amount of the bill by the value of$l in francs: the quotient is the amount to be paid in dollars. 2. What will be the amount to be paid, United States money, for a bill of exchange on Paris, of 6530 francs, exchange being 5.14 francs per dollar ? 3. What will be the amount to be paid in United States money for a bill of exchange on Paris of 10262 francs, ex- change being 5.09 francs per dollar ? 4. What will be the value in United States money of a bill for 87595 francs, at 5.16 francs per dollar? DUTIES. 277. Persons who bring goods or merchandise into the United States, from foreign countries, are required to land them at particular places or Ports, called Ports of Entry, and to pay a certain amount of their value, called a Duty. This duty is imposed by the General Government, and must be the same on the same articles of merchandise, in every part of the United States. Besides the duties on merchandise, vessels employed in commerce are required, by law, to pay certain sums for the privilege of entering the ports. These sums are large or small, in proportion to the size or tonnage of the vessels. The moneys arising from duties and tonnage, are called revenues. 276. What is the unit or base of the French currency ? What is its value? How is it divided ? In what currency arc French bills of ex- change drawn ? 277. What is a port of entry? What is a duty? By whom are duties imposed ? What charges are vessels required to pay ? What are the moneys arising from duties and tonnage called ? DUTIES. 269 278. The revenues of the country are under the general direction of the Secretary of the Treasury, and to secure their faithful collection, the government has appointed various officers at each port of entry or place where goods may be landed. 279. The office established by the government at any port of entry is called a Custom House, and the officers attached to it are called Custom House Officers. 280. All duties levied by law on goods imported into the United States, are collected at the various custom houses, and are of two kinds, Specific and Ad valorem. A specific duty is a certain sum on a particular kind of goods named ; as so much per square yard on cotton or wool- len cloths, so much per ton weight on iron, or so much per gallon on molasses. An ad valorem duty is such a per cent on the actual cost of the goods in the country from which they are imported. Thus, an ad valorem duty of 15 per cent on English cloth, is a duty of 15 per cent on the cost of cloths imported from Eng- land. 281. The laws of Congress provide, that the cargoes of all vessels freighted with foreign goods or merchandise shall be weighed or gauged by the custom house officers at the port to which they are consigned. As duties are only to be paid on the articles, and not on the boxes, casks and bags which con- tain them, certain deductions are made from the weights and measures, called Allowances. Gross Weight is the whole weight of the goods, together with that of the hogshead, barrel, box, bag, &c., which con- tains them. L ; __^____ 278. Under whose direction are the revenues of the country ? 279. What is a custom house ? What are the officers attached to it called ? 280. Where are the duties collected ? How many kinds are there, and what are they called ? What is a specific duty ? An ad valorem duty ? 281. What do the laws of Congress direct in relation to foreign goods? Why are deductions made from their weight? What are these deductions called ? What is gross weight ? What is draft ? What is the greatest draft allowed ? ' What is tare ? What arc the different kinds of tare ? What allowances are made on liquors ? 270 DUTIES. Draft is an allowance from the gross weight on account of waste, where there is not actual tare. On 112/6. it is 1/6. From 112 to 224 < 2, 224 to 336 ' 3, 336 to 1120 ' 4, 1120 to 2016 ' 7, Above 2016 any weight ' 9 ; consequently, 9/6. is the greatest draft allowed. Tare is an allowance made for the weight of the boxes, barrels, or bags containing the commodity, and is of three kinds : 1st, Legal tare, or such as is established by law ; 2d, Customary tare, or such as is established by the custom among merchants ; and 3c?, Actual tare, or such as is found by re- moving the goods and actually weighing the boxes or casks in which they are contained. On liquors in casks, customary tare is sometimes allowed on the supposition that the cask is not full, or what is called its actual wants; and then an allowance of 5 per cent for leakage. A tare of 10 per cent is allowed on porter, ale and beer, in bottles, on account of breakage, and 5 per cent on all other liquors in bottles. At the custom house, bottles of the com- mon size are estimated to contain 2J gallons the dozen. NOTE. For table8 of Tare and Duty, see Ogden on the Tariff of 1842. EXAMPLES. 1. What will be the duty on 125 cartons of ribbons, each containing 48 pieces, and each piece weighing 802. net, and paying a duty of $2.50 per pound ? 2. What will be the duty on 225 bags of coffee, each weigh- ing: gross 160/6., invoiced at 6 cents per pound ; 2 per cent being the legal rate of tare, and 20 per cent the duty ? 3. What duty must be paid on 275 dozen bottles of claret, estimated to contain 2J gallons per dozen, 5 per cent beinar allowed for breakage, and the duty being 35 cents per gallon? 4. A merchant/ imports 175 cases of indigo, each case weighing 196/fo?. gross ; 15 per cent is the customary rate of tare, and the duty 5 cents per pound : what duty must he pay on the whole ? ALLIGATION MEDIAL. 271 ALLIGATION MEDIAL. 282. ALLIGATION MEDIAL is the process of finding the price of a mixture when the quantity of each simple and its price are known. 1. A merchant mixes Sib. of tea, worth 75 cents a pound, with 16/6. worth $1.02 a pound : what is the price of the mixture per pound ? ANALYSIS. The quantity, 8lb. of OPERATION. tea, at 75 cents a pound, costs $6 ; 8/6. at 75cte.=$ 6 00 and 16. at $1.03 costs $16.32 : 16/6 at $1 Q2 = $16.32 hence, the mixture, = 24lb,, costs \ . $22.32 ; and the price of lib. of the 24 24)22.32 mixture is found by dividing this $0 93 cost by 24 : hence, to find the price of the mixture, I. Find the cost of the entire mixture : II. Divide the entire cost of the mixture by the sum of the simples, and the quotient will be the price of the mixture. EXAMPLES. 1. A farmer mixes 30 bushels of wheat worth 5s. per bushel, with 72 bushels of rye at 3s. per bushel, and with 60 bushels of barley worth 2s. per bushel : what should be the price of a bushel of the mixture ? 2. A wine merchant mixes 15 gallons of wine at $1 per gallon with 25 gallons of brandy worth 75 ceuts per gallon : what should be the price of a gallon of the compound ? 3. A grocer mixes 40 gallons of whisky worth 31 cents per gallon with 3 gallons of water which costs nothing : what should be the price of a gallon of the mixture ? 4. A goldsmith melts together 2/6. of gold of 22 carats fine, 602:. of 20 carats fine, and 6oz. of 16 carats fine : what is the fineness of the mixture ? 5. On a certain day the mercury in the thermometer was observed to average the following heights : from 6 in the morning to 9, 64 ; from 9 to 12, 74 ; from 12 to 3, 84 ; and from 3 to 6, 70 : what was the mean temperature of the day ? 282. What is Alligation Medial ? What is the rule for determining the price of the mixture ? 272 ALLIGATION ALTERNATE. ALLIGATION ALTERNATE. 283. ALLIGATION ALTERNATE is the process of finding what proportions must be taken of each of several simples, whose prices are known, to form a compound of a given price. It is the opposite of Alligation Medial, and may be proved by it. 284. To find the proportional parfe. 1. A farmer would mix oats at 3s. a bushel, rye at 6s., and wheat at 9s. a bushel, so that the mixture shall be worth 5 shillings a bushel : what proportion must be taken of each sort? OPERATION, oats, 3 5 -j rye, wheat, 9 A. B. c. D. E. 2 1 3 2 2 1 1 ANALYSIS. On every bushel put into the mixture, whose price is less than the mean price, there will be a gain ; on every bushel whose price is greater than the mean price, there will be a loss ; and since there is to be neither gain nor loss by the mixture, the gains and losses must balance each other. A bushel of oats, when put into the mixture, will bring 5 shil- lings, giving a gain of 2 shillings ; and to gain 1 shilling, we must take half as much, or \ a bushel, which we write in column A. On 1 bushel of wheat there will be a loss of 4 shillings ; and to make a loss of 1 shilling, we must take of a bushel, which we also write in column A : i and are called proportional numbers. Again : comparing the oats and rye, there is a gain of 2 shil- lings on every bushel of oats, and a loss of 1 shilling on every bushel of rye : to gain 1 shilling on the oats, we take \ a bushel, and to lose 1 shilling on the rye, we take 1 bushel : these num- bers are written in column B. Two simples, thus compared, are called a couplet : in one, the price of unity is less tJian the mean price, and in the other it is greater. If, every time we take i a bushel of oats we take ^ of a bushel of wheat, the gain and loss will balance ; and if every time we take ^ a bushel of oats we take 1 bushel of rye, the gain and loss 283. What is Alligation Alternate ? J284. How do you lind the proportional numbers/* ALLIGATION ALTERNATE. 273 will balance : hence, if tTie proportional numbers of a couplet be multiplied by any number, the gain and loss denoted by the products, will balance. When the proportional numbers, in any column, are fractional (as in columns A and B), multiply them by the least common multiple of their denominators, and write the products in new columns C and D. Then, add the numbers in columns C and D, standing opposite each simple, and if their sums have a common factor, reject it : the last result Will be the proportional numbers. RULE. I. Write the prices or qualities of the simples in a column, beginning with the lowest, and the mean price or quality at the left. II. Opposite the first simple write the part which must be taken to gain 1 of the mean price, and opposite the other simple of the couplet, write the part which must be taken to lose 1 of the mean price, and do the same for each simple. III, W hen the proportional numbers are fractional, reduce them to integral numbers, and then add those which stand oppo- site the same single: if the sums have a common factor, reject it : the result will denote the proportional parts. 2. A merchant would mix wines worth 16s., 18s., and 22s. per gallon, in such a way, that the mixture may be worth 20s. per gallon : what are the proportional parts ? OPERATION. . A. B. C. D. E. (161 204l8 J (22 1 1 1 i 1 1 1 1 1 3 PROOF. 1 gallon, at 16 shillings, == 16s. 1 gallon, at 18 shillings, = 18s. 3 gallon, at 22 shillings, = 66s. 5) 100 (2 Os., mean price. N'OTE. The answers to the last, and to all similar questions, will be infinite in number, for two reasons: 1st. If the proportional numbers in column E be multiplied by any number, integral or fractional, the products will denote pro- portional parts of the simples. 2d. If the proportional numbers of any couplet be multiplied by 18 274: ALLIGATION ALTERNATE. any number, the gain and loss in that couplet will still balance, and the proportional numbers in the final result will be changed. 3. What proportions of tea, at 24 cents, 30 cents, 33 cents and 36 cents a pound, must be mixed together so that the mixture shall be worth 32 cents a pound ? 4. What proportions of coffee at IQcts., 20cts. and 28cfe. per pound, must be mixed together so that the compound shall be worth 24ds. per pound ? 5. A goldsmith has gold of 16, of 18, of 23, and of 24 carats fine : what part must be taken of each so that the mixture shall be 21 carats fine? 6. What portion of brandy, at 14s. per gallon, of old Ma- deira, at 24s per gallon, of new Madeira, at 21s. per gallon, and of brandy, at 10s. per gallon, must be mixed together so that the mixture shall be worth 18s. per gallon ? 285. When the quantity of one simple is given : I. How much wheat, at 9s. a bushel, must be mixed with 20 bushels of oats worth 3 shillings a bushel, that the mix- ture may be worth 5 shillings a bushel ? ANALYSIS. Find the proportional numbers : they are 2 and 1 ; hence, the ratio of the oats to the wheat is \ : therefore, there, must be 10 bushels of wheat. RULE. I. Find the proportional numbers, and write the given single opposite its proportional number. II. Multiply the given simple by the ratio which its propor- tional number bears to each of the others, and the products will denote the quantities to be taken of each. EXAMPLES. 1. How much wine, at 5s., at 5s. Gd., and 6s. per gallon must be mixed with 4 gallons, at 4s. per gallon, so that the mixture shall be worth 5s. 4d. per gallon ? 2. A fanner would mix 14 bushels of wheat, at $1,20 per bushel, with rye at 72c/s., barley at 48cs., and oats at 36c/s. : how much must be taken of each sort to make the mixture worth 64 cents per bushel ? 3. There is a mixture made of wheat at 4s. per bushel, rye at 3s., barley at 2s., with 12 bushels of oats at l&d. per bushel : how much is taken of each sort when the mixture is worth 3s. Qd. ? ALLIGATION ALTERNATE. 275 4. A distiller would mix 40^ro/. of French brandy at 12s. per gallon, with English at Is. and spirits at 4s. per gallon : what quantity must be taken of each sort that the mixture may be afforded at 8s. per gallon ? 286. When the quantity of the mixture is given. 1. A merchant would make up a cask of wine containing 50 gallons, with wine worth 16s., 18s. and 22s. a gallon, in such a way that the mixture may be worth 20s. a gallon : much must he take of each sort ? ANALYSIS. This is the same as example 2, except that the quantity of the mixture is given. If the quantity of the mixture be divided by 5, the sum of the proportional parts, the quotient 10 will show how many times each pwportional part must be taken to make up 50 gallons : hence, there are 10 gallons of the first, 10 of the second, and 30 of the third : hence, RULE. I. Find the proportional parts. II. Divide the quantity of the mixture by the sum of the proportional parts, and the quotient will denote how many times each part is to be taken. Multiply this quotient by the parts separately, and each product will denote the quan- tity of the corresponding simple. EXAMPLES. 1. A grocer has four sorts of sugar, worth 12c?., Wd., 6d and 4:d. per pound ; he would make a mixture of 144 pounds worth Sd. per pound : what quantity must be taken of each sort? 2. A grocer having four sorts of tea, worth 5s., 6s., 8s. and 9s. per pound, wishes a mixture of 87 pounds worth 7s, per pound : how much must he take of each sort ? 3. A silversmith has four sorts of gold, viz., of 24 carats fine, of 22 carats fine, and of 20 carats fine, and of 15 carats fine : he would make a mixture of 42oz. of 17 carats fine ; how much must be taken of each sort ? PROOF. All the examples of Alligation Medial may be found by Alligation Alternate. 285. How do you find the quantity of each simple when the quantity of one simple is known ? 386. How do you find the quantity of each simple when the quantity of each mixture is known ? 276 INVOLUTION. INVOLUTION. 287. A POWER is the product of equal factors. The equal factor is called the root of the power. The first power is the equal factor itself, or the root : The second power is the product of the root by itself : The third power is the product when the root is taken 3 times as a factor : The fourth power, when it is taken 4 times : The fifth power, when it is taken 5 times, &c. 288. The number denoting how many times the root is taken as a factor, is called the exponent of the power. It is written a little at the right and over the root : thus, if the equal factor or root is 4, 4= 4 the 1st power of 4. 4 2 4x4= 16 the 2d power of 4. 43 _4 X 4 X 4 64 the 3d power of 4. 4 4 =4: x 4 x 4 x 4 = 256 the 4th power of 4. 45 .-4x4x4x4x4 1024 the 5th power of 4. INVOLUTION is the process of finding the powers of 'number 's. NOTES. 1. There are three things connected with every power : 1st, The root ; 2d, The exponent ; and 3d, The power or result of the multiplication. 2 In finding a power, the root is always the 1st power; hence, the number of multiplications is 1 less than the exponent; RULE. Multiply the number by itself as many times less 1 as there are units in the exponent, and the last product will be the power. EXAMPLES. Find the powers of tne following numbers : 1. Square of 1. 2. Square of J. 3. Cube of |. 4. Square of f . 5. Square of 9. 6. Cube of 12 1. 3d power of 125. 8. 3d power of 16 9. 4th power of 9. 10. 5th power of 16. 11. 6th power of 20. 12. 2d power of 225 13. Square of 2167. 14. Cube of 321 15. 4th power of 215. 16. 5th power of 906. 17. 6th power of 9. 18. Square of 36049. EVOLUTION. 277 EVOLUTION. 289, EVOLUTION is the process of finding the factor when we know the power. The square root of a number is the factor which multiplied by itself once will produce the number. The cube root of a number is the factor which multiplied by itself twice will produce the number. Thus, 6 is the square root of 36, because 6 x 6=36 ; and 3 is the cube root of 27, because 3 x 3 x 3=27. The sign V is called the radical sign. When placed be- fore a number it denotes that its square root is to be ex- tracted. Thus, 1/36 = 6. We denote the cube root by the same sign by writing 3 over it : thus, v^ denotes the cube root of 27, which is equal to 3. The small figure 3, placed over the radical, is called the index of the root. 'EXTRACTION OF THE SQUARE ROOT. 290. The square root of a number is a factor which mul- tiplied by itself once will produce the number. To extract the square root is to find this factor* The first ten numbers and their squares are 1, 2, 3, 4, 5, 6, Y, 8, 9, 10. 1, 4, 9, 16, 25, 36, 49, 64, 81. 100. The numbers in the first line are the square roots of those in the second. The numbers 1, 4, 9, 16, 25, 36, &c. having exact factors, are called perfect squares. A perfect square is a number which has two exact factors NOTE. The square root of a number less than 100 will be less than 10, while the square root of a number greater than 100 will be greater than 10. 287. What is a power ? What is the root of a power? What is the first power ? What is the second power ? The third power ? 288. What is the exponent of the power ? How is it written ? What is Involution ? How many things are connected with every power ? How do you find the power of a number ? 289. What is Evolution? What is the square root of a number? What is the cube root of a number ? How do you denote the square root of a number ? How the cube root ? 278 EXTRACTION OF THE SQUARE ROOT. 291. What is the square of 36=3 tens + 6 units? ANALYSIS. 36=3 tens+6 units, is first to be taken 6 units' time, giving 6 2 +3 x 6 : then taking it 3 tens' times, we have 3 x 6+3 2 , and the sum is 3 2 +2(3 x 6)+6 2 : that is, 3 + 6 3 + 6 3x6 + 6* 3 2 +3x6 3 2 +2(3x6)+6 The square of a number is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units. The same may be shown by the figure : Let the line AB re- F 30 present the 3 tens or 30, and BC the six units. Let AD be a square on AC, and AE a square on the ten's line AB. Then ED will be a square on the unit line 6, and the rectangle EF will be the product of HE, which is equal to the ten's line, by IE, which is equal to the unit line Also, the rectangle BK will be the product of EB, which is equal to the ten's line, by the unit line B C. But the whole square on AC is made up of the square AE, the two rectangles FE and EC, and the square ED. 1. Let it now be required to extract the square root of 1296. ANALYSIS. Since the number contains more than two places of figures, its root will contain tens and units. But as the square of one ten is one hundred, it follows that the square of the tens of the required root must be found in the two figures on the left of 96. Hence, we point off the number into periods of two figures each. 30 6 180 6 6 36 30 E 900 + 180 + 180 + 36=1296. 30 30 30 6 900 180 30 C 290. What is the square root of a, number ? What are perfect squares ? How many are there between 1 and 100 ? 291. Into what parts may a number be decomposed? When so de- composed, what is its square equal to ? EXTRACTION OF THE SQUARE ROOT. 279 We next find the greatest square contained in OPERATION. 12, which is 3 tens or 30. We then square 3 1296(36 tens which gives 9 hundred, and then place 9 un- ~ der the hundreds' place, and subtract , this takes away the square of the tens, and leaves 396, 66)396 which is twice the product of the tens by the units 395 plus the square of the units. If now, we double the divisor and then divide this remainder, exclusive of the right hand figure, (since that figure cannot enter into the product of the tens by the units) by it, the quotient will be the units figure of the root. If we annex this figure to the augmented divisor, and then multiply the whole divisor thus in- creased by it, the product will be twice the tens by the units plus the square of the units ; and hence, we have found both figures of the root. This process may also be illustrated by the figure. Subtracting the square of the tens is taking away the square AE and leaves the two rectangles FE and BK, together with the Bquare ED on the unit line. The two rectangles FE and BK*representing the product of units by tens, can be expressed by no figures less than tens. If, then, we divide the figures 39, at the left of 6, by twice the tens, that is, by twice AB or BE, the quotient will be BG or EK the unit of the root. Then, placing BC or G, in the root, and also annexing it to the divisor doubled, and then multiplying the whole divisor 66 by 6, we obtain the two rectangles FE and CE, together with the equare ED. 292. Hence, for the extraction of the square root, we have the following RULE. I. Separate the given number into periods of two figures each, by setting a dot over the place of units, a se- cond over the place of hundreds, and so on for each alternate figure at the left. II. Note the greatest square contained in the period on the left, and place its root on the right after the manner of a quotient in division. Subtract the square of this root from the first period, and to the remainder bring down the second period for a dividend. 292. What is the first step in extracting the square root of numbers ? What is the second? What is the third? What the fourth? What the fifth ? Give the entire rule. 280 EXTRACTION OF THE SQUARE ROOT. III. Double the root thus found for a trial divisor and place it on the left of the dividend. Find how many times the trial divisor is contained in the dividend, exclu- sive of the right-hand figure, and place the quotient in the root and also annex it to the divisor. IY. Multiply the divisor thus increased, by the last figure of the root ; subtract the product from the dividend, and to the remainder bring down the next period for a new divi- dend. Y. Double the ivhole root thus found, for a new trial di- visor, and continue the operation as before, until all the periods are brought down. EXAMPLES. 1. What is the square root of 263169 ? OPERATION. ANALYSIS. We first place a dot over the a o'i A 6 / K i Q 9, making the right-hand period 69. We then put a dot over the 1 and also over the 6, making three periods. 101)131 The greatest perfect square in 26 is 25, AI the root of which is 5, Placing 5 in the root, subtracting its square from 26, and 1023)3069 bringing down the next period 31, we have 3069 131 for a dividend, and by doubling the root we have 10 for a trial divisor. Now, 10 is contained in 13, 1 time. Place 1 both in the root and in the divisor : then multi- ply 101 by 1 ; subtract the product and bring down the next period. We must now double the whole root 51 for a new trial divisor ; or we may take the first divisor after having doubled the last figure 1 ; then dividing, we obtain 3, the third figure of the root. NOTE. 1. The left-hand period may contain but one figure; each of the others will contain two. 2. If any trial divisor is greater than its dividend, the corres ponding quotient figure will be a cipher. 3. If the product of the divisor by any figure of the root exceeds the corresponding dividend, the quotient figure is too large and must be diminished. 4. There will be as many figures in the root as there are periods in the given number. 5. If the given number is not a perfect square there will be a remainder after all the periods are brought down. In this case, periods of ciphers may be annexed, forming new periods, each of which will give one decimal place in the root. EXTRACTION OF THE SQUARE ROOT. 281 What is the square root of 36729 : OPERATION. 3 67 29(191.64 + 1 In this example there are two periods of decimals, which give two places of decimals in the root. 29)267 261 381)629 381 3826)24800 22956 38324)184400 153296 31104 Hem. 293. To extract the square root of a fraction. 1. What is the square root of .5 ? NOTE. We first annex one cipher to make even decimal places. We then ex- tract the root of the first period : to the remainder we annex two ciphers, forming a new period, and so on. OPERATION. .50(.707 + 49 140)100 000 1407)10000 9849 151 Rem. OPERATION. 2. What is the square root of ? NOTE. The square root of a fraction is equal to the square root of the numerator divided by the square root of the denomi- nator. 3. What is the square root of J ? NOTE. When the terms are not per- fect squares, reduce the common fraction | = . 7 5 ; to a decimal fraction, and then extract x /sZr v /VcT_ the square root of the decimal. 5 *& OPERATION. 293. How do you extract the square root of a decimal fraction ? ef a common fraction ? How 282 SQUARE ROOT. RULE. I. If ike fraction is a decimal, point off the periods from the decimal point to the right, annexing ci- phers if necessary, so that each period shall contain two places, and then extractJhe root as in integral numbers. II. If the fraction is a common fraction, and its terms perfect squares, extract the square root of the numerator and denominator separately ; if they are not perfect squares, re- duce the fraction to a decimal, and then extract the square root of the result. EXAMPLES. What are the square roots of the following numbers ? of 3? of 11? of 1069 ? of 2268741? 5. of 7596796? of 36372961? of 22071204? of 3271.4207? of 4795.25731? 10. of 4.372594? 11. of .0025? 12. of .00032754? 13. of .00103041? 14. of 4.426816? 15. of8f ? 16. of 9J? 17. of^? 18. o 19. o 20. off APPLICATIONS IN SQUARE ROOT. 294. A triangle is a plain figure which has three sides and three angles. If a straight line meets another straight line, making the adjacent angles equal, each is called a right angle ; and the lines are said to be perpendicular to each other. 295. A right angled triangle is one which has one right angle. In the right angled triangle ABC, the side AC opposite the right angle B is called the hi/pothenuse ; the side AB the base; and the side BC the perpendicular. APPLICATIONS. 283 29G. In a right angled triangle the square described in the hypothemise is equal to the sum of the squares described in the other two sides. Thus, if AC13 be a right angled triangle, right an- gled at C, then will the large square, D, described on the hypothenuse AB, be 1 equal to the sum of the squares F and E described on the sides AC and CB. This is called the carpen- ter's theorem. By count- ing the small squares in the large square D, you will find their number equal to that contained in the small squares F and E. In this triangle the hypothenuse AB = 5, AC = 4, and CB = 3. Any numbers having the same ratie, as 5, 4 and 3, such as 10, 8 and 6 ; 20, 16 and 12, &c., will represent the sides of a right angled triangle. 1. Wishing to know the distance from A to the top of a tower, I measured the height of the tower and found it to be 40 feet ; also the distance from A to B and found it 30 feet ; what was the distance from A to C ? 30 2 = 900 BC=40; BC^40 2 ^ ~ 2500 = ^2500 = 50 feet. 297. Hence, when the base and perpendicular are known and the hypothenuse is required, 294. What is a triangle ? What is a right angle ? 295. What is a right angled triangle ? Which side is the hypothe- nuse ? 296. In a right angled triangle what is the square on the hypothe- nuse equal to ? 284 SQUARE ROOT. Square the base and square the perpendicular, add the re- sults and then extract the square root of their sum. 2. What is the length of a rafter that will reach from the eaves to the ridge pole of a house, when the height of the roof is 15 feet and the width of the building 40 feet ? 298. To find one side when we know the hypothenuse and the other side. 3. The length of a ladder which will reach from the mid- dle of a street 80 feet wide to the eves of a house, is 50 feet : what is the height of the house ? Ans. 30 feet. ANALYSIS Since the square of the length of the ladder is equal to the sum of the squares of half the street and the height of the house, the square of the length of the ladder diminished by the square of half the street will be equal to the square of the height of the house : hence, Square the hypothenuse and the known side, and take the difference ; the square root of the difference will be the other side. EXAMPLES. 1. If an acre of land be laid out in a square form, what will be the length of each side in rods ? 2. What will be the length of the side of a square, in rods, that shall contain 100 acres ? 3. A general has an army of 7225 men : how many must be put in each line in order to place them in a square form ? 4. Two persons start from the same point ; one travels due east 50 miles, the other due south 84 miles : how far are they apart ? 5. What is the length, in rods, of one side of a square that shall contain 12 acres ? 6. A company of speculators bought a tract of land for $6724, each agreeing to pay as many dollars as there were partners : how many partners were there ? 297. How do you find the hypothenuse when you know the base and perpendicular ? 298. If you know the hypothenuse and one side, how do you find the other side ? CUBE ROOT. 285 7. A farmer wishes to set out an orchard of 3844 trees, so that the number of rows shall be equal to the number of trees in each row : what will be the number of trees ? 8. How many rods of fence will enclose a square field of 10 acres ? 9. If a line 150 feet long will reach from the top of a steeple 120 feet high, to the opposite side of the street, what is the width of the street ? 10. What is the length of a brace whose ends are each 3| feet from the angle made by the post and beam ? CUBE ROOT. 299. The CUBE ROOT of a number is one of three equal factors of the number. To extract the cube root of a number is to find a factor which multiplied into itself twice, will produce the given number. Thus, 2 is the cube root of 8 ; for, 2 x 2 x 2 = 8 : and 3 is the cube toot of 27 ; for 3 x 3 x 3 = 27. 1, 2, 3, 4, 5, 6, 7, 8, 9. 1 8 27 64 125 216 343 512 729. The numbers in the first line are the cube roots of the corresponding numbers of the second. The numbers of the second line are called perfect cubes. By examining the num- bers of the two lines we see, 1st. That the cube of units cannot give a higher order than hundreds. 2d. That since the cube of one ten (10) is 1000 and the cube of 9 tens (90), 81000, the cube of tens will not give a lower denomination than thousands, nor a higher denomi- nation than hundreds of thousands. Hence, if a number contains more than three figures, its cube root will contain more than one : if it contains more than six, its root will contain more than two, and so on ; every additional three figures giving one additional figure in the root, and the figures which remain at the left hand, although less than three, will also give a figure in the root, This law explains the reason for pointing off into periods of three figures each. 286 CUBE BOOT. 300. Let us now see how the cube of any number, as 16, is formed. Sixteen is composed of 1 ten and 6 units, and may be written 10 -f G. To nod the cube of 16, or of 10+6, we must multiply the number by itself twice To do this we place the number thus 16=10-}- 6 10+ 6 product by the units - 60+36 product by the tens -100+ 60 Square of 16 - 100+ 120--*- 36 Multiply again by 16 - - 10+6 product by the units - 600+ 720+216 product by the tens 1000+1200+ 360 Cube of 1 6 TOOO+T800 + 1080 + 2l6 1. By examining the parts of this number it is seen that the first part 1000 is the cube of the tens ; that is, 10x10x10=1000. 2. The second part 1800 is three times the square of the tens multiplied by the units ; that is, 3 x (10)* x 6=3 x 100 x 6=1800. 3. The third part 1080 is three times the square of the units multiplied by the tens ; that is, 3 x6 2 x 10=3x36x10=1080. 4. The" fourth part is the cube of the units ; that is, 6 3 =6x 6x6=210. 1. What is the cube root of the number 4096 ? ANALYSTS. Since the number contains more than three figures, 4 096(16 we iuaow that the root will con- 1 tain at least units and tens. ia o \1T~n 7c\ Q T R . Separating the three right- l*X_3 = o)3 I hand figures from the 4, we 16 3 =4 096 know that the cube of the tens \vili be found in the 4 ; and 1 is the greatest cube in 4. 299. What is the cube root of a number ? How many perfect cubes arc there between 1 and 1000 ? Tin,.* 800. Of how many parts is the cube of a number composed ? What are they ? CUBE BOOT. 287 Hence, we place the root 1 on the right, and this is the tens of the required root. We then cube 1 and subtract the result from 4, and to the remainder we bring down the first figure of the next period. We have seen that the second part of the cube of 16, viz. 1800, is three times the square of the tens multiplied by the units : and hence, it can have no significant figure of a less denomination than hundreds. It must, therefore, make up a part of the 30 hundreds above. But this 30 hundreds also contains all the hundreds which come from the 3d and 4th parts of the cube of 16. If it were not so, the 30 hundreds, divided by three times the square of the tens, would give the unit figure exactly Forming a divisor of three times the square of the tens, we find the quotient to be ten , but this we know to be too large. Placing 9 in the root and cubing 19, we find the result to be 6859. Then trying 8 we find the cube of 18 still too large ; but when we take 6 we find the exact number. Hence the cube root of 4096 is 16. 301. Hence, to find the cube root of a number, RULE. I. Separate the given number into periods of three figures each, by placing a dot over the place of units, a second over the place of thousands, and so on over each third figure to the left ; the left hand period will often contain less than three places of figures. IT. Note the greatest perfect cube in the first period, and set its root on the right, after the manner of a quotient in di- vision. Subtract the cube of this n umber from the first period, and to the remainder bring down the first figure of the next period for a dividend. III. Take three times the square of the root just found for a trial divisor, and see how often it is contained in the divi- dend, and place the quotient for a second figure of the root. Then cube the figures of the root thus found, and if their cube be greater than the first two periods of the given num- ber, diminish the last figure, but if it be less, subtract it from the first two periods, and to the remainder bringdown the first figure of the next period for a new dividend. IY. Take three times the square of the whole root for a second trial divisor, and find a third figure of the root. Cube the whole root thus found and subtract the result from the first three periods of the given number when it is less than that number, but if it is greater, diminish the figure of the root / proceed in a similar way for all the periods. 288 CUBE ROOT. EXAMPLES. 1. What is the cube root of 99252841 ? 99 252 847(463 4 3 =64 4? x 3=48)352 dividend. First two periods 99 252 (46)*=46x 46x46= 97 336 3 x (46) 2 =634S ) 19T68 2d dividend. The first three periods - 99 252 847 (463) 3 =99 252 847 Find the cube roots of the following numbers : 1. Of 389017? 2. Of 5735339? 3. Of 32461759? 4. Of 84604519? 5. Of 259694072? 6. Of 48228544? 302. To extract the cube root of a decimal fraction. Annex ciphers to the decimal, if necessary, so that it shall consist of 3, 6, 9, &c., places. Then put the first point over the place of thousandths, the second over the place of millionths, and so on over every third place to the right ; after which extract the root as in whole numbers. NOTES. 1. There will be as many decimal places in the root as there are periods in the given number. 2. The same rule applies when the given number is composed of a whole number and a decimal. 3. If in extracting the root of a number there is a remainder after all the periods have been brought down, periods of ciphers may be annexed by considering them as decimals. EXAMPLES. Find the cube roots of the following numbers.: 1. Of .157464? 2. Of .870983875 ? 3. Of 12.977875? 4. Of .751089429? f>. Of .353393243 ? 6. Of 3.408862625? 301. What is the rule for extracting the cube root ? 303. How do you extract the cube root of a decimal fraction ? How many decimal places will there be in the root ? Will the same rulft apply when there is a whole number and a decimal ? If in extracting the root of any number you find i decimal, how do you proceed ? APPLICATIONS. 289 303. To extract the cube root of a common fraction. I. Reduce compound fractions to simple ones, mixed num- bers to improper fractions, and then reduce the fraction to its lowest terms. II. Extract the cube root of the numerator and denomi- nator separately, if they have exact roots ; but if either of them has not an exact root, reduce the fraction to a decimal and extract the root as in the last case, EXAMPLES. Find the cube roots of the following fractions : 1. Offf|? 4. Of? 2. Of31J&? 5. Off? 3- Of T 3^? 6. Of |? APPLICATIONS. 1. What must be the length, depth, and breadth of a box, when these dimensions are all equal and the box contains 4913 cubic feet ? 2. The solidity of a cubical block is 21952 cubic yards : what is the length of each side ? What is the area of the surface ? 3. A cellar is 25 feet long 20 feet wide, and 8| feet deep : what will be the dimensions of another cellar of equal capacity in the form of a cube ? 4. What will be the length of one side of a cubical granary that shall contain 2500 bushels of grain ? 5. How many small cubes of 2 inches on a side can be sawed out of a cube 2 feet on a side, if nothing is lost in sawing ? 6. What will be the side of a cube that shall be equal to the contents of a stick of timber containing 1728 cubic feet? 7. A stick of timber is 54 feet long and 2 feet square : what would be its dimensions if it had the form of a cube ? NOTES. 1. Bodies are said to be similar when their like parts are proportional. 2. It is found that the contents of similar bodies are to each other as the cubes of their like dimensions. 303. How do you extract the cube root of a vulgar fraction ? 19 290 ARITHMETICAL PROGRESSION, 3, All bodies named in the examples are supposed to be simi lar. 8. If a sphere of 4 feet in diameter contains 33.5104 cubic feet, what will be the contents of a sphere 8 feet in diameter ? 4 3 : 8 3 : : 33.5104 : Am. 9. If the contents of a sphere 14 inches in diameter is 1436.7584 cubic inches, what will be the diameter of a sphere which contains 11494.0672 cubic inches ? 10. If a ball weighing 32 pounds is 6 inches in diameter, what will be the diameter of a ball weighing 2048 pounds ? 11. If a haystack, 24 feet in height, contains 8 tons of hay, what will be the height of a similar stack that shall contain but 1 ton ? ARITHMETICAL PROGRESSION. 304. An Arithmetical Progression is a series of numbers in which each is derived from the preceding one by the addition or subtraction of the same number. The number added or subtracted is called the common dif- ference. 305. If the common difference is added, the series is called an increasing series. Thus, if we begin with 2, and add the common difference, 3, we have 2, 5, 8, 11, 14, 17, 20, 23, &c., which is an increasing series. If we begin with 23, and subtract the common difference, 3, we hare 23, 20, 17, 14, 11, 8, 5, &c., which is a decreasing series. 304. What is an arithmetical progression ? What is the number added or subtracted called? 305. When the common difference is added, what is the scries called ? What is it called when the common difference is subtracted ? What are the several numebrs called ? What arc the first and last called ? What arc the intermediate ones called ? ARITHMETICAL PROGRESSION. 291 The several numbers are called the terms of the progres- sion or series : the first and last are called the extremes, and the intermediate terms are called means. 306. In every arithmetical progression there are five parts : 1st, the first term ; 2d, the last term ; 3d, the common difference ; 4th, the number of terms ; 5th, the sum of all the terms. If any three of these parts are known or given, the remain- ing ones can be determined. CASE I. 307. Knowing the first term, the common difference, and the number of terms, to find the last term. 1. The first term is 3, the common difference 2, and the number of terms 19 : what is the last term ? ANALYSIS. By considering the manner in which the increasing progression is formed, we see that the 2d term is obtained by adding the common difference to the 1st term ; the 3d, by OPEBATION. adding the common difference to the 2d ; the 1 8 No. less 1 4th, by adding the common difference to the cj Com dif 3d, and so on ; the number of additions being 1 less than the number of terms found. 35 But instead of making the additions, we may 3 1st term, multiply the common difference by the number ^7: , , , of additions, that is, by 1 less than the number m of terms, and add the first term to the pro- duct : hence, RULE. Multiply the common difference by 1 less than the number of terms ; if the progression is increasing, add the product to the first term and the sum ivill be the last term ; if it is decreasing, subtract the product from the first term and the difference will be the la?t term. 306. How many parts are there in every arithmetical progression ? What are they ? How many parts must be given before the remaining ones can be found ? 292 ARITHMETICAL PROGRESSION. EXAMPLES. 1. A man bought 50 yards of cloth, for which he was tQ pay 6 cents for the 1st yard, 9 cents for the 2d, 12 cents for the 3d, and so on increasing by the common difference 3 : how much did he pay for the last yard ? 2. A man puts out $100 at simple interest, at 1 per cent : at the end of the 1st year it will have increased to $107, at the end of the 2d year to $114, and so on, increasing $t each year : what will be the amount at the end of 1 6 years ? 3. What is the 40th term of an arithmetical progression of which the first term is 1, and the common difference 1 ? 4. What is the 30th term of a descending progression of which the first term is 60, and the common difference 2 ? 5. A person had 35 children and grandchildren, and it so happened that the difference of their ages was 18 months, and the age of the eldest was 60 years : how old was the youngest ? CASE II. 308. Knowing the two extremes and the number of terms, to find the common difference. 1. The extremes of an arithmetical progression are 8 and 104, and the number of terms 25 : what is the common dif- ference ? ANALYSIS. Since the common difference multiplied by 1 less than the number of OPERATION. terms gives a product equal to the differ 104 erence of the extremes, if we divide the dif g ference of the extremes by 1 less than the number of terms, the quotient will be the 25 1 24)96(4. common difference : hence, RULE. Subtract the less extreme from the greater and divide the remainder by 1 less than the number of terms; the quotient will be the common difference. 307. "When you know the first term, the common difference, and the number of terms, how do you find the last term ? 308. When you know the extremes and the number of terms, how do you find the common difference ? ARITHMETICAL PROGRESSION. 293 EXAMPLES. 1. A man has 8 sons, the youngest is 4 years old and the eldest 32 : their ages increase in arithmetical progression : what is the common difference of their ages ? 2. A man is to travel from New York to a certain place in 12 days ; to go 3 miles the first day, increasing every day by the same number of miles,; the last day's journey is 58 miles : required the daily increase. 3. A man hired a workman for a month of 26 working days, and agreed to pay him 50 cents for the first day, with a uniform daily increase ; on the last day he paid $1.50 : what was the daily increase ? CASE III. 309. To find the sum of the terms of an arithmetical progression. 1. What is the sum of the series whose first term is 3, common difference 2, and last term 19 ? Given scries - 3+ 5 + 1 + 9 + 11 + 13 + 15 + 17 + 19= 99 ofTcnnshv-l 19 + 17 + 15 + 13 + 11+ 9+ t+ 5+ 8= 99 verted. J Sura of both. 2'2 iJ 22 22 22 22 22 22 22 198 ANALYSIS. The two series are the same ; hence, their sum is equal to twice the given series. But their sum is equal to the sum of the two extremes 3 and 19 taken as many times as there are terms ; and the given series is equal to half this sum, or to the sum of the extremes multiplied by half the number of terms. RULE. Add the extremes together and multiply their sum by half the number of terms ; the product will be the sum of the series. EXAMPLES. 1. The extremes are 2 and 100, and the number of terms 22 : what is the sum of the series? OPERATION. ANALYSIS. We first add 2 1st term, together the two extremes inn lost tpvm and then multiply by half la the number of terms. 1 02 sum of extremes. 11 half the number of terms 1122 sum of series. 309. How do you find the sum of the terms? 294 GEOMETRICAL PKOGEESSION. 2. How many strokes does the hammer of a clock strike iu 12 hours? 3. The first term of a series is 2, the common difference 4, end the number of terms 9 : what is the last term and sum of the series ? 4. James, a smart chap, having learned arithmetical pro- gression, told his father that he would chop a load of wood of 15 logs, at 2 cents for the first log, with a regular increase of 1 cent for each additional log : how much did James receive for chopping the wood ? 5. An invalid wishes to gain strength by regular and in- creasing exercise ; his physician assures him that he can walk 1 mile the first day, and increase the distance half a mile for each of the 24 following days : how far will he walk ? C. If 100 eggs are placed in a right line, exactly one yard from each other, and the first one yard from a basket : what distance will a man travel who gathers them up singlv and places them in the basket ? GEOMETRICAL PROGRESSION. 310. A GEOMETRICAL PROGRESSION is a series of terms, each of which is derived from the preceding one, by multi- plying it by a constant number. The constant multiplier is called the ratio of the progression. 311. If the ratio is greater than 1, each term is greater than the preceding one, and the series is said to be in- creasing. 31.0. What is a geometrical progression? What is the constant multiplier called ? 311. If the ratio is greater than 1, how do the terms compare with each other? What is the series then called? If the ratio is less than 1, how do they compare ? What is the series then called ? What arc the several numbers called? What are the first and last called? What are the intermediate ones called ? 312. How many parts are there in every geometrical progression ? What are they? How manv must be known before the others can be found ? GEOMETRICAL PROGRESSION. 295 If the ratio is less than 1, each term is less than the preceding one, and the series is said to be decreasing; thus, 1, 2, 4, 8, 16 ; 32, &c. ratio 2 increasing series : 32, 16, 8, 4, 2, 1, &c. ratio 1 decreasing series. The several numbers are .called terms of the progression. The first and last are called the extremes, and the intermedi- ate terms are called means. 312. In every Geometrical, as well as in every Arithmeti- cal Progression, there are five parts : 1st, the first term ; 2d, the last term ; 3d, the common ratio ? 4th, the number of terms ; 5th, the sum of all the terms. If any three of these parts are known, or given, the re- maining ones can be determined. CASE I. 313. Having given the first term, the ratio, and the number of terms, to find the last term. 1. The first term is 3 and the ratio 2 : what is the 6th term? ANALYSIS. The se- OPERATION. cond term is formed by 2x2x2x2x 2=:2 5 = 32 multiplying the first 3 j t t term by the ratio ; tho _____ third term by multiply- Ans. 96 ing the second term by the ratio, and so on ; the number of multiplications being 1 less iJian the number of terms : thus, 3 3 1st term, 3x2 = 6 2d term, * 3x2x2=3x2-=12 3d term, 3 x 2 x 2 x 2^3 x 2 3 24 4th term, <fcc. 296 GEOMETRICAL PROGRESSION. Therefore, the last term is equal to the first term multi- plied by the ratio raised to a power 1 less than the number of terms. RULE. Eaise the ratio to a power whose exponent is 1 less than the number of terms, and then multiply this power by the first term. EXAMPLES. 1. The first term of a decreasing progression is 192 ; the ratio i, and the number of terms 7 : what is the last term ? NOTE. The 6th power of the ratio, (-), is OPERATION. ^4, and this multiplied by the first term 192, (l) 6 = Jk- gives the last term 3. 1 92 X -^-=3 2. A man purchased 12 pears ; he was to pay 1 farthing for the 1st, 2 farthings for the 2d, 4 for the 3d, and so on, doubling each time : what did he pay for the last ? 3. The first term of a decreasing progression is 1024, the ratio i : what is the 9th term ? 4. The first term of an increasing progression is 4, and the common ratio 3 : what is the 10th term ? 5. A gentleman dying left nine sons, and bequeathed his estate in the following manner : to his executors $50 ; his youngest son to have twice as much as the executors, and each son to have double the amount of the son next younger : what was the eldest son's portion ? 6. A man bought 12 yards of cloth, giving 3 cents for the 1st yard, 6 for the 2d, 12 for the 3d, &c. : what did he pay for the last yard ? CASE II. 314. Knowing the two extremes and the ratio, to find the sum of the terms. 1. What is the sum of the terms in the progression, 1, 4, 16, 64 ? 313. Knowing the first term, the ratio, :ind the number- of terms, 1 row- do you find the Itust term ? 314. Knowing the two extremes and the ratio, how do you find the sum of the terms V GEOMETRICAL PROGRESSION. 297 ANALYSIS. If we multiply the terms of the progression by the Tatio 4, we have a second pro- gression, 4, 16, 64, 256, which OPERATION. is 4 times as great as the first. 4+16+64+256= 4 times. If from this we subtract the 1+4+16+64 =_ once. first, the remainder, 2561, 256 1=3 times. will be 3 times as great as 9 ~/ 1 9 ~,- the first; and it the remain- !==-- = 85 sum. der be divided by 3, the quo- ' tient will be the sum of the terms of the first progression. But 256 is the product of the last term of the given progression multiplied by the ratio, 1 is the first term, and the divisor 3 is 1 less than the ratio ; hence, RULE. Multiply the last term by the ratio ; take the dif- ference between the product and the first term and divide the remainder by the difference between 1 and the ratio. NOTE. When the progression is increasing, the first term is subtracted from the product of the last term by the ratio, and the divisor is found by subtracting 1 from the ratio. When the pro- gression is decreasing, the product of the last term by the ratio is subtracted from the 'first term, and the ratio is subtracted from 1. EXAMPLES. 1. The first term of a progression is 2, the ratio 3, ami the last term 4374 : what is the sum of the terms ? 2. The first term of a progression is 128, the ratio J, and the last term 2 : what is the sum of the terms ? 3. The first term is 3, the ratio 2, and the last term 192 : what is the sum of the series ? 4. A gentleman gave his daughter in marriage on New Year's day, and gave her husband Is. towards her portion, and was to double it on the first day of every month during the year : what was her portion ? 5. A man bought 10 bushels of wheat*on the condition that he should pay 1 cent for the 1st bushel, 3 for the 2d, 9 for the 3d, and so on to the last : what did he pay for the last bushel, and for the 10 bushels? 6. A man has 6 children : to the 1st he gives $150, to the 2d $300, to the 3d $600, and so on, to each twice as much as the last : how much did the ehKst, Teceive, and what was the amount received by them all ? 298 PROMISCUOUS QUESTIONS. PROMISCUOUS EXAMPLES. 1. A merchant bought 13 packages of goods, for which he paid $326 : what will 39 packages cost at the same rate ? 2. How many bushels of oats at 62^ cents a bushel will pay for 4250 feet of lumber at $7.50 per thousand ? 3. Bougkt 27ihd. of sugar which weighed as follows : the 1st 5cwt. Iqr. ISlb., the 2d Gcwt. IQlb. : what did it cost at 7 cents per pound? 4. How many hours between the 4th of Sept., 1854, at 3 P.M., and the 20th day of ApriJ, 1855, at 10 A.M. ? 5. If | of a gallon of wine cost of a dollar, what will - of a hogshead cost ? 6. What number is that which being multiplied by \ will pro- duce i? 7. A tailor had a piece of cloth containing 24 yards, from which he cut 6 1 yards : how much was there left ? 8. From | offtake lof^' 9. What is the difference between 3| + 7| and 4 + 2-H ? 10. There was a company of soldiers, of whom \ were on guard, preparing dinner, and the remainder, 85 men, were drilling : ow many were there in the company ? 11. The sum of two numbers is 425, and their difference 1.625: what are the numbers ? 12. The sum of two numbers is f, and their difference ^ : what are the numbers ? 13. The product of two numbers is 2.26, and one of the numbers is .25 : what is the other ? 14. If the divisor of a certain number be 6.66, and the quo- tient \ , what will be the dividend ? 15. A person dying, divided his property between his widow and his four sons ; to his widow he gave $1780, and to each of his sons $1250 ; he had been 25^ years in business, and had cleared on an average 126 dollars a year : how much had he when he began business ? 16. A besieged garrison consisting of 360 men was provisioned for 6 months, but hearing of no relief at the end of five months, dismissed so many of the garrison, that the remaining provision lasted 5 months : how many men were sent away ? 17. Two persons, A and B are indebted to C ; A owes $2173, which is the least debt, and the difference of the debts is $371 : what is the amount of their indebtedness ? 18. What number added to the 43d part of 4429 will make the sum 240 ? PROMISCUOUS QUESTIONS. 299 19. How many planks 15 feet long, and 15 inches wide, will floor a barn 60^ feet long, and 33i feet wide? 20. A person owned f of a mine, and sold f of his interest for $ 1710 : what was the value of the entire mine ? 21. A room 30 feet long, and 18 feet wide, is to be covered with painted cloth f of a yard wide : how many yards will cover it ? 22. A, B and C trade together and gain $120, which is to be shared according to each one's stock ; A put in $140, B $300, and C $160 : what is each man's share. 23. A can do a piece of work in 12 days, and B can do the same work in 18 days : how long will it take both, if they work together? 24. If a barrel of flour will last one family 7 months, a second family 9 months, and a third ll months, how long will it last the "three families together ? 25. Suppose I have -,% of a ship worth $1200 ; what part have I left after selling | of $ of my share, and what is it worth? 26. What number is that which being multiplied by of f of 1 , the product will be 1 ? 27. Divide $420 between three persons, so that the second shall have f as much as the first, and the third ^ as much as the other two ? 28. What is the difference between twice five and fifty, and twice fifty five ? 29. What number is that which being multiplied by three- thousandths, the product will be 2637 ? 30. What is the difference between half a dozen dozens and six dozen dozens? 31. The slow or parade step is 70 paces per minute, at 28 inches each pace : how fast is that per hour ? 32. A lady being asked her age, and not wishing to give a direct answer, said, " I have 9 children, and three years elapsed between the birth of each of them ; the eldest was born when I was 19 years old, and the youngest is now exactly 19 :" what was her age ? 33. A wall of 700 yards in length was to be built in 29 days : 12 men were employed on it for 11 days, and only completed 220 yards : how many men must be added to complete the wall in the required time ? 34. Divide $10429.50 between three persons, so that as often as one gets $4, the second will get $6 and the third $7. 35. A gentleman whose annual income is 1500, spends 20 guineas a week ; does he save, or run in debt, and how much ? 36. A farmer exchanged 70 bushels of rye, at $0.92 per bushel, for 40 bushels of wheat, at $1.874/ a bushel, and received the balance in oats, at $0.40 per bushel : how many bushels of oats did he receive ? 37. In a certain orchard of the trees bear apples, i of them bear peaches, of them plums, 120 of them cherries, and 80 of them pears: how many trees are there in the orchard ? 300 PKOMISCUOUS QUESTIONS. 38. A person being asked the time, said, the time past noon is equal to of the time past midnight : what was the hour ? 89. If 20 men can perform a piece of work in 12 days, how many men will accomplish thrice as much in one-fifth of the time? 40. How many stones 2 feet long, 1 foot wide, and 6 inches thick, will build a wall 12 yards long, 2 yards high, and 4 feet thick ? 41. Four persons traded together on a capital of $6000, of which A put in , B put in ^, C put in %, and D the rest ; at the end of 4 years they had gained $4728 : what was each one's share of the gain ? 42. A cistern containing 60 gallons of water has three unequal pipes for discharging it ; the largest will empty it in one hour, the second in two hours, and the third in three hours : in what time will the cistern be emptied if they run together ? 43. A man bought f of the capital of a cotton factory at par ; he retained of his purchase, and sold the balance for $5000 which was 15 per cent advance on the cost ; what was the whole capital of the factory ? 44. Bought a cow for $30 cash, and sold her for $35 at a credit of 8 months : reckoning the interest at 6 per cent, how much did I gain ? 45. If, when I sell cloth for 8-?. Qd. per yard, I gain 12 per cent, what per cent will be gained when it is sold for 10s. Qd per yard ? 46. How much stock at par value can be purchased for $8500, at 8^ per cent premium, per cent being paid to the broker? 47. Twelve workmen, working 12 hours a day, have made in 12 days, 12 pieces of cloth, each piece 75 yards long ; how many pieces of the same stuff would have been made, each piece 25 yards long, if there had been 7 more workmen ? 48. A person was born on the 1st day of Oct., 1801, at 6 o'clock in the morning, what was his age on the 21st of Sept., 1854, at half-past 4 in the afternoon? 49. A, can do a piece of work alone in 10 days, and B in 13 days : in what time can they do it if they work together? 50. A man went to sea at 17 years of age; 8 years after he had a son born, who lived 46 years, and died before his father ; after which the father lived twice twenty years and died : what was the age of the father ? 51. How many bricks, 8 inches long and 4 inches wide, will pave a yard that is 100 feet by 50 feet ? 52. If a house is 50 feet wide, and the post which supports the ridge pole is 12 feet high, what will be the length of the rafters? 53. A man had 12 sons, the youngest was 3 years old and the eldest 58, and their ages increased in Arithmetical progression: what was the common difference of their ages ? PROMISCUOUS QUESTIONS. 301 54. If a quantity of provisions serves 1500 men 12 weeks, at the rate of 20 ounces a day for each man, how many men will the same provisions maintain for 20 weeks, at the rate of 8 ounces a day for each man ? 55. A man bought 10 bushels of wheat, on the condition that he should pay 1 cent for the 1st bushel, 3 for the 3d, 9 for the 3d, and so on to the last : what did he pay for the last bushel, and for the 10 bushels ? 56. There is a mixture made of wheat at 4s. per bushel, rye at 3s., barley at 2s., with 12 bushels of oats at 18d. per bushel : how much must be taken of each sort to make the mixture worth 2s., tid. per bushel ? 57. What length must be cut off a board 8^ inches broad to contain a square foot ? 58. What is the difference between the interest of $2500 for 4 years 9 mo. at 6 per cent., and half that sum for twice the time, at half the same rate per cent ? 59. A person lent a certain sum at 4 per cent, per annum ; had this remained at intera Bt 3 years, he would have received for prin- cipal and interest $9676.80 : what was the principal? 60. If: 1 pound of tea be equal in value to 50 oranges, and 70 oranges be worth 84 lemons, what is the value of a pound of tea, when a lemon is worth 2 cents ? 61. A person bought 160 oranges at 2 for a penny, and 180 more at 3 for a penny ; after which he sold them out at the rate of 5 for 2 pence .did he make or lose, and how much ? 62. A snail in getting up a pole 20 feet high, was observed to climb up 8 feet every day, but to descend 4 feet every night : in what time did he reach the top of the pole ? 63. A ship has a leak by which it would fill and sink in 15 hours, but by means of a pump it could be emptied, if full, in 16 hours. Now, if. the pump is worked from the time the leak begins, how long before the ship will sink ? 64. A and B can perform a certain piece of work in 6 days, B and C in 7 days, and A and C in 14 days : in what time would each do it alone ? 65. Divide $500 among 4 persons, so that when A has i dollar B shall have , C, |, and D . 66. A man purchased a building lot containing 3600 square feet, at the cost of $1.50 per foot, on which he built a store at an expense of $3000. He paid yearly $180.66 for repairs and taxes : what annual rent must he receive to obtain 10 per cent on the cost? 67. A's note of $7851.04 was dated Sept. 5th, 1837, on which were endorsed the following payments, viz. : Nov. 13th, 1839, $416.98; May 10th, 1840, $152- what was due March 1st, 1841, the interest being 6 per cent ? 302 PROMISCUOUS QUESTIONS. 68. A Louse is 40 feet from the ground to the caves, and it is required to find the length of a ladder which will reach the eaves, supposing the foot of the ladder cannot be placed nearer to the house than 30 feet ? G9. Sound travels about 1142 feet in a second ; now, if the flash of a cannon be seen at the moment it is fired, and the report heard 45 seconds after, what distance would the observer be from the gun ? 70. A person dying, worth $5460, left a wife and 2 children, a son and daughter, absent in a foreign country. He directed that " if his son returned, the mother should have one third of the estate and the son the remainder ; but if the daughter returned, she should have one third, and the mother the remainder. Now it so happened that they both returned : how mustthe estate be divided to fulfill the father's intentions ? 71. Two persons depart from the same place, one travels 82, and the other 36 miles a day : if they travel in the same direction, how far will they be apart at the end of 19 days, and how far if they travel in contrary directions ? 72. In what time will $2377.50 amount to $2852.42, at 4 per cent, per annum ? 73. What is the height of a wall, which is 14^ yards in length, and -fo of a yard in thickness, and which has cost $406, it having been paid for at the rate of $10 per cubic yard ? 74. What will be the duty on 225 bags of coffee, each weighing gross 160 Ibs., invoiced at 6 cents per Ib. ; 2 per cent, being the legal rate of tare, and 20 per cent, the duty ? 75. Three persons purchase a piece of property for $9202 ; the first gave a certain Bum ; the second three times as much ; and the third one and a half time as much as the other two: what did each pay ? 76. A reservoir of water has two pipes to supply it. The first would fill it in 40 minutes, and the second in 50. It has likewise a discharging pipe, by which it may be emptied when full in 25 minutes. Now, if all the pipes are opened at once, and the water runs uniformly as we have supposed, how long before the cistern will be filled? 77. A traveller leaves New Haven at 8 o'clock on Monday morning, and walks towards Albany at the rate of 3 miles an hour : another traveller sets out from Albany at 4 o'clock on the same evening, and walks towards New Haven at the rate of 4 miles an hour ; now, supposing the distance to be 130 miles, where on the road will they meet ? MENSURATION. 303 MENSURATION. 315. A triangle is a portion of a plane bounded by three straight lines. BC is called the base ; and AD, perpendicular to BC, the altitude. 316. To find the area of a triangle. The area or contents of a triangle is equal to the product of half its base by its altitude (Bk. IV. Prop. VI).* EXAMPLES. 1. The base, BC, of a triangle is 40 yards, and the perpendicu- lar, AD, 20 yards ; what is the area ? 2. In a triangular field the base is 40 chains, and the perpendi- cular 15 chains : how much does it contain ? (ART. 110.) 3. There is a triangular field, of which the base is 35 rods and the perpendicular 26 rods : what are its contents ? 317. A? square is a figure having four equal sides, and all its angles right angles. 318. A rectangle is a four-sided figure like a square, in which the sides are perpendicular to each other, but the adjacent sides are not equal. 319. A parallelogram is a four-sided figure which has its opposite sides equal and parallel, but its angles not right angles. The line DE, perpendi- cular to the base, is called the altitude. 320. To find the area of a square, rectangle, or parallelogram, Multiply the base by the perpendicular height, and the product will be the area. (Book IV. Prop. V). EXAMPLES. 1. What is the area of a square field of which the sides are each 33.08 chains ? 2. What is the area of a square piece of land of which the sides are 27 chains? 3. What is the area of a square piece of land of which the sides are 25 rods each ? * All the references arc to Davies' Legendre. 304: MENSURATION. 4. What are the contents of a rectangular field, the length of which is 40 rods and the breadth 20 rods ? 5. What are the contents of a field 40 rods square ? 6. What are the contents of a rectangular field 15 chains long and 5 chains broad ? 7. What are the contents of a field 27 chains long and 9 rods broad ? 8. The base of a parallelogram is 271 yards, and the perpendi. cular height 360 feet : what is the area ? 321. A trapezoid is a four-sided figure ABCD, having two of its sides, AB, DC, parallel. The perpendicular CE is called the altitude. 322. To find the area of a trapezoid. Multiply half the sum of the two parallel sides "by the alti- tude, and the product will be the area. (Bk. IV. Prop. VII.) EXAMPLES. 1. Required the area of the trapezoid ABCD, having given AB=321.51/., DC=214.24/*., and CE=171.16/^. 2. What is the area of a trapezoid, the parallel sides of which are 12.41 and 8.22 chains, and the perpendicular distance between them 5.15 chains '? 3. Required the area of a trapezoid whose parallel sides are 25 feet 6 inches, and 18 feet 9 inches, and the perpendicular distance between them 10 feet and 5 inches. 4. Required the area of a trapezoid whose parallel sides are 20.5 and 12.25, and the perpendicular distance between them 10.75 yards. 5. What is the area of a trapezoid whose parallel sides are 7.50 chains, and 12.25 chains, and the perpendicular height 15.40 chains V 6. What are the contents when the parallel sides are 20 and 32 chains, and the perpendicular distance between them 26 chains ? 323. A circle is a portion of a plane bounded by a curved line, called the circum- ference. Every point of the circumference is equally distant from a certain point within called the centre : thus, C is the centre, and any line, as ACB, passing through the centre, is called a diameter. If the diameter of a circle' is 1, the circumference will be 3.1416. Hence, if we know the diameter, ^ce may find the circum- ference by multiplying by 3.1416 ; or, if we know the circumference., we may find the diameter by dividing by 3.1416. MENSURATION. 305 EXAMPLES. 1. The diameter of a circle is 4, what is the circumference ? 2. The diameter of a circle is 93, what is the circumference ? 3. The diameter of a circle is 20, what is the circumference ? 4. What is diameter of a circle whose circumference is 78.54 ? 5 What is the diameter of a circte whose circumference is 11052.1944? 6. What is the diameter of a circle whose circumference is 6850 ? 324. To find the area or contents of a circle. Multiply the square of the diameter by the decimal .7854 (Bk. V. Prop. XII. Cor. 2). EXAMPLES. 1. What is the area of a circle whose diameter is 6 ? 2. What is the area of a circle whose diameter is 10? 3. What is the area of a circle whose diameter is 7 ? 4. How many square yards in a circle whose diameter is 3i feet ? 325. A sphere is a figure terminated by a curved surface, ull the parts of which are equally distant from a certain point within called the centre. The line AB passing through its centre C is called the diameter of the sphere, and AC its radius. o~6. To find the surface of a sphere, Multiply the square of the diameter by 3.1416 (Bk. VIII. Prop. X. Cor). EXAMPLES. 1. What is the surface of a sphere whose diameter is 12 ? 2. What is the surface of a sphere whose diameter is 7 ? 3. Required the number of square inches in the surface of a sphere whose diameter is 2 feet or 24 inches. 327. To find the contents of a sphere, Multiply the surface by the diameter and divide the product by 6; the quotient mil be the contents. (Bk. VIII. Prop. XIV. Sch. 3.) EXAMPLES 1. What are the contents of a sphere whose diameter is 12 ? 2. What are the contents of a sphere whose diameter is 4 ? 3. What are the contents of a sphere whose diameter is 14i7i. ? 4 What are the contents of a sphere whose diameter is Gfl. ? 20 306 MENSURATION. 328. A prism is a figure whose ends are equal plane figures and whose faces are paralelograms. The sum of the sides which bound the base is called the perimeter of the base, and the sum of the parallelograms which bound the solid is called the convex surface. 329. To find the convex surface of a right prism, Multiply the perimeter of the base by the perpendicular height, and thegtroduct will be the convex surface (Bk. VII. Prop. I). EXAMPLES. 1. What is the convex surface of a prism whose base is bounded by five equal sides, each of which is 35 feet, the altitude being 26 feet? 2. What is the convex surface when there are eight equal sides, each 15 feet in length, and the altitude is 12 feet ? 330. To find the solid contents of a prism. Multiply the area of the base by the altitude, and the product will be the contents (Bk. VII. Prop. XIV). EXAMPLES. 1. What are the contents of a square prism, each side of the square which forms the base being 15, and the altitude of the prism 20 feet ? 2. What are the contents of a cube each side of which is 24 inches ? 3. How many cubic feet in a block of marble of which the length is 3 feet 2 inches, breadth 2 feet 8 inches and height or thickness 2 feet 6 inches ? 4. How many gallons of water will a cistern contain whose di- mensions are the same as in the last example ? 5. Required the contents of a triangular prism whose height is 10 feet, and area of the base 350 ? 331. A cylinder is a figure with circular ends. The line EF is called the axis or alti- tude, and the circular surface the convex sur- face of the cylinder. MENSURATION. 307 332. To find the convex surface, Multiply the circumference of the base by the altitude, and the product ivill be the convex surface. (Bk. VIII. Prop. I.) EXAMPLES. 1 What is the convex surface of a cylinder, the diameter of whose base is 20 and the altitude 50 ? 2. What is the convex surfa'ce of a cylinder, whose altitude is 14 feet and the circumference of its base 8 feet 4 inches ? 3. What is the convex surface of a cylinder, the diameter of whose base is 30 inches and altitude 5 feet ? 333. To find the contents of a cylinder, Multiply the area of the base by the altitude : the product will be the contents. (Bk. VIII. Prop. II). EXAMPLES. 1. Required the contents of a cylinder of which the altitude is 12 feet and the diameter of the base 15 feet ? 2. What are the contents of a cylinder, the diameter of whoso base is 20 and the altitude 29? 3. What are the contents of a cylinder, the diameter of whose base is 12 and the altitude 30 ? 4. What are the contents of a cylinder, the diameter of whose base is 16 and altitude 9 ? 5. What are the contents of a cylinder, the diameter of whose base is 50 and altitude 15 ? 334. A pyramid is a figure formed by several triangular planes united at the same point S, and terminating in the different sides of a plain figure as ABCDE. The altitude of the pyramid is the line SO, drawn perpendicular to the base. 335. To find the contents of a pyramid, Multiply the area of the base by one-third of the altitude. (Bk. VII, Prop XVII). 308 MENSUEATlQJS'. EXAMPLES. 1. Required the contents of a pyramid, of which the area of the base is 95 and the altitude 15. 2 What are the contents of a pyramid, the area of whose base is 260 and the altitude 24 ? 3. What are the contents of a pyramid, the area of whose base is 207 and altitude 18? 4 What are the contents of a pyramid, the area of whose base is 403 and altitude 30 ? 5. What are the contents of a pyramid, the area of whose base is 270 and altitude 16? 6. A pyramid has a rectangular base, the sides of which are 25 and 12 : the altitude of the pyramid is 36 : what are its con- tents ? 7. A pyramid with a square base, of which each side is 30, has an altitude of 20 : what are its contents ? 336. A cone is a figure with a circular base, and tapering to a point called the vertex. The point C is the vertex, and the line CD is called the axis or altitude. 337. To find the contents of a cone, Multiply the area of the base ly one-third of tJie altitude. (Bk. VIII. Prop. V.) EXAMPLES. 1. Required the contents of a cone, the diameter of whose base is 5 and the altitude 10. 2. What are the contents of a cone, the diameter of whose base is 18 and the altitude 27 ? 3. What are the contents of a cone, the diameter of whose base is 20 and the altitude 30 ? 4. What are the contents of a cone, whose altitude is 27 feet and the diameter of the base 10 feet ? 5. What are the contents of a cone, whose altitude is 12 feet and the diameter of its base 15 feet ? GAUGING 309 GAUGING-. 338. The mean diameter of a cask is found by adding to tho head diameter, two thirds of the difference between the bung and head diameters, or if the staves are not much curved, by adding" six-tenths. This reduces the cask to a cylinder. Then, to find the solidity, we multiply the square of the mean diameter by the decimal .7854 and the product by the length. This will give the solid contents in cubic inches. Then, if we divide by 231, we have the contents in gallons. (Art. 114). Multiply the length by the square of the OPERATION. mean diameter, then by the decimal .7854, Ixd 2 X '- 7 - 8 w 5 -^- and divide by 231. ' J x d 2 x .0034. If, then, we divide the decimal ,7854 by 231, the quotient car- ried to four places of decimals is .0034, and this decimal multi- plied by the square of the mean diameter and by the length of the cask, will give the contents in gallons. 339. Hence, for gauging or measuring casks, we have the fol. lowing HULE. Multiply the length "by the square of the mean diameter ; then multiply ly 34 and point off four decimal places, and the pro- duct icill then express gallons and the decimals of a gallon. 1. How many gallons in a cask whose bung diameter is 36 inches, head diameter- 30 inches, and length 50 inches ? We first find the difference of the diameters, OPERATION. of which we take two thirds and add to the 3630= 6 head diameter. We then multiply the square 2 O f 6 = 4 of the mean diameter, the length and 34 3 Q()4-4. 4 together, and point off four decimal places in the product. 34 =1156 2. What is the number of gallons in a -. Qr r 7 cask whose bung diameter is 38 inches, head lyo.O^roc. diameter 32 inches, and length 42 inches ? 3. How many gallons in a cask whose length is 36 inches, bung diameter 35 inches, and head diameter 30 inches ? 4. How many gallons in a cask whose length is 40 inches, head diameter 34 inches, and bung diameter 38 inches? 5 A water tub holds 147 gallons ; the pipe usually brings in 14 gallons in 9 minutes : the tap discharges at a medium, 40 gal- Jons in 31 minutes. Now, supposing these to be left open, and the water to be turned on at 2 o'clock in the morning ; a servant at 5 shuts the tap, and is solicitous to know at what time the tub will be filled in case the water continues to flow. 310 APPENDIX, FORMS RELATING TO BUSINESS IN GENERAL. FORMS OF OKDERS. MESSRS. M. JAMES & Co. Please pay John Thompson, or order, five hundred dollars, and place the same to my account, for value received. PETER WORTHY. Wilmington, N. 0., June 1, 1855. MR. JOSEPH RICH, Please pay, for value received, the bearer, sixty-one dollars and twenty cents, in goods from your store, and charge the same to the account of your Obedient Servant JOHN PARSONS. Savannafi, Ga., July 1, 1855. FORMS OF RECEIPTS. Receipt for Money on account. Received, Natchez, June 2d, 1855, of John Ward, sixty dollars on account. $60,00 JOHN P. FAY. Receipt for Money on a Note. Received, Nashville, June 5, 1856, of Leonard Walsh, six hun- dred and forty dollars, on his note for one thousand dollars, dated New York, January 1, 1855. $640,00 J. N. WEEKS. NOTES. 1. A NOTE, or as it is generally called, a promissory note, is a positive engagement, in writing, to pay a given sum at a time specified, either to a person named in the note, or to his order, or to the bearer. 2. By mercantile usage a note does not really fall due until the expiration of 3 days after the time mentioned on its face. The three additional days are called days of grace. APPENDIX. 311 When the last day of grace happens to be Sunday, or a holiday, such as New Years, or the Fourth of July, the note must be paid the day before : that is, on the second day of grace. 3. There are two kinds of notes discounted at banks : 1st. Notes given by one individual to another for property actually sold these are called business notes, or business paper. 3d. Notes made for the purpose of borrowing money, which are called accommo- dation notes, or accommodation paper. Notes of the first class are much preferred by the banks, as more likely to be paid when they fall due, or in mercantile phrase, " when they come to maturity." FORMS OP NOTES. No. 1. Negotiable Note. $25,50. . Providence, May 1, 1856. For value received I promise to pay on demand, to Abel Bond, or order, twenty-five dollars and 50 cents. REUBEN HOLMES. Note Payable to Bearer. No. 2. $875,39. St. Louis, May 1, 1855. For value received I promise to pay, six months after date, to John Johns, or bearer, eight hundred and seventy-five dollars and thirty-nine cents. PIERCE PENNY. Note by two Persons. No. 3. $659,27. Buffalo, June 2, 1856. For value received we, jointly and severally, promise to pay to Richard Ricks, or order, on demand sis hundred and fifty- nine dollars and twenty-seven cents. ENOS ALLAN. JOHN ALLAN. Note Payable at a Bank. $20,25. Chicago, May 7, 1856. Sixty days after date, I promise to pay John Anderson, or order, at the Bank of Commerce in the city of New York, twenty dollars and twenty-five cents, for value received. JESSE STOKES. 312 APPENDIX. REMARKS RELATING TO NOTES. 1. The person who signs a note, is called the drawer or maker of the note ; thus, Reuben Holmes is the drawer of Note No. 1. 2. The person who has the rightful possession of a note, is called the holder of the note. 3. A note is said to be negotiable when it is made payable to A B, or order, who is called the payee (see No. 1). Now, if Abel Bond, to whom this note is made payable, writes his name on the back of it, he is said to endorse the note, and he is called the en- dorser ; and when the note becomes due, the holder must first demand payment of the maker, Reuben Holmes, and if he declines paying it, the holder may then require payment of Abel Bond, the endorser. 4 If the note is made payable to A B, or bearer, then the drawer alone is responsible, and he must pay to any person who holds the note. 5. The time at which a note is to be paid should always be named, but if no time is specified, the drawer must pay when re- quired to do so, and the note will draw interest after the payment is demanded. 6. When a note, payable at a future day, becomes due, and is not paid, it will draw interest, though no mention is made of inter- est. 7. In each of the States there is a rate of interest established by law, which is called the legal interest, and when no rate is speci- fied, the note will always draw legal interest. If a rate higher than legal interest be taken, the drawer, in most of the States, is not bound to pay the note. 8. In the State of New York, although the legal interest is 7 per cent, yet the banks are not allowed to charge over G per cent, unless the notes have over 63 days to run. 9. If two persons jointly and severally give their note, (see No. 3,) it may be collected of either of them. 10. The words "For value received" should bo expressed in every note. 11. When a note is given, payable on a fixed day, and in a spe- cific article, as in wheat or rye, payment must be offered at the specified time, and if it is not, the holder can demand the value in money. A BOND FOR ONE PERSON, WITH A CONDITION. KNOW ALL MEN BY THESE PRESENTS, THAT I, James Wilson of the City of Hartford and State of Connecticut, am held and firmly bound unto John Pickens of the Town of Waterbury, County of New Haven and State of Connecticut, in the sum of APPENDIX. 313 Eighty dollars lawful money of the United States of America, to be paid to the said John Pickens, his executors, administrators, or assigns : for which payment well and truly to be made J bind myself, my heirs, executors, and administrators, firmly by these presents. Sealed with my Seal. Dated the Ninth day of March, one thousand eight hundred and thirty-eight. THE CONDITION of the above obligation is such, that if tlio above bounden James Wilson, his heirs, executors, or administra- tors, shall well and truly pay or cause to be paid, unto the above- named John Pickens, his executors, administrators, or assigns, the just and full sum of [Here insert the condition.] then the above obligation to be void, otherwise to remain in full force and virtue. Sealed and delivered in the presence of John Frost, ) Joseph Wiggins,) James Wilson. NOTE. The part in Italic to be filled up according to circum- stance. If there is no condition to the bond, then all to be omitted after and including the words, " THE CONDITION, &c." BOOK-KEEPING. PERSONS transacting business find it necessary to wiite down the articles bought or sold, together with their prices and the names of the persons to whom sold. BOOK-KEEPING is the method of recording such transactions in a regular manner. COMMON ACCOUNT BOOK. The following is a very convenient form for book-keeping, and requires but a single book. l"c is probably the best form of a com- mon Account Book. J. BELL. DR J. BELL. CR. 1846. $ c. 1846. *|. June 1 11 6 July 9 To 5 cords of wood, at $1,75 per cord, To 1 day's work, To 4bn. of rye, at 62 cents per bu. 8 1 2 75 00 48 July ( " 1C u 20 Aug. 1 By shoeing horse, u mending sleigh, " ironing wagon, " Cash to balance, 100 325 512 386 12 23 12i23 314 ANSWERS. p. EX. ANS. EX. ANS EX ANS. EX. ANS. 24. 24. 9 10 577 7689 11 12 502616 799999 13 14 43 cts. 73 cts. 15 |888 20. 25. 17 18 4083 6846 19 20 9798 8601 21 22 7032 979 23 559 2Ve 27. 27. 27. 5 6 7 8 12089 26901 28637 203933 9 10 11 12 23272 - 233642 247481 1994439 13 14 15 16 175874 172775 98967 10742750 28. 28. 28. 20 21 22 787676921 100570011 15371781930 23 24 25 26754 730528 7047897 2fc 27 25687540 297303078 29. 29. 29. 28 29 30 13115375 3942805S 140700034 31 32 33 1819857171537 1105354 1079167 34 1118969 30.|| 1 365 || 2 5567 ||3| 16375||4|421||5|392||6 34671660 31. 31. 31. 7 8 9 82869 2576406 270 10 11 4596-119* J4239052< ( 453090S ) 12 ) 13 > 1287462 1665400 32. 32. 32. 14 15 16 50994 143985 2728116 17 18 19 5990267 6644374 7685134 20 21 23191876 23191876 37. 37. 37. 37. 9 10 11 12 260822 2935621 50391719 28443 13 14 15 16 99246591 999999 776462 18561747 17 18 19 4244083 8013105 52528 38. 38. 38. 1 2 3 10 - 45 $1115 4 23^ 5 5 6 t f 7 8 9 62 785608 37 10 11 12 175502 696 2687 39. 39. 39. 39. 13 14 15 16 250-$1500 26 1860805 17 18 19 20 239 1759 55 21 22 23 24 190 $4020-1340 2769818 94 ANSWERS. 315 p. EX. ANS. EX. ANS. EX. ANS EX. ANS. 40. 40. 25 26 145 168 27 28 168 137 29 30 15914260 20463760 31 2769818 40.|| 1 | 29045 | 2 $418 ! 3 | $714 4 | $5795 41. 5 $390 fc ! $919 11 230-527 14 11854617 41. ^ $224980 { ) 55 12 19553068 41. 7 $1706 1C ) 28223 13 $3818 47. 9 936 11 $298 13 $28511 _ 47. 10 $1236 12 35688 14 $6578 49. 3 7913576 12 65948806 21 764819895290424 49. 4 2537682 13 36914176 22 6241519790 49. 5 4280822 14 85950000 23 105062176 49. 6 19014604 15 3320863272 24 601380780 49. 7 85564584 16 816515040 25 4984155396 49. 8 2183178497 17 68959488 26 405768300 49. 9 93939864472 18 35843685 27 800105244 49. 10 395061696 19 267293339604 28 1227697160 49. 11 393916488 20 214007086881 29 330445150 51. 2 274032 4 15076944 6 .7430778 51. 3 19180896 5 50618898 7 553248 52. 52. 52. 52. 52. 52. 1 2 3 4 5 6 2540 64800 7987000 98400000 375000 67040000 7 8 1 2 3 4 214100 87200000 1833600 4368560000 148512000000 1315170000000 5 6 7 8 9 25 196 10 521 9175000000 0310474010 1484000 9215040000 0018850000 53. | 1|480||2|4415||3|168||4 $291 || 5| 2214-123 || 6 | 11680 54. 54. 54. 54. 54. 7 8 9 10 11 3087 18755 119568 984072 24427326 12 13 14 15 16 349440 1057500 150000 131250000 53095 17 18 19 20 16 4 20-2220 968710 2720 08-2040 55. 55. 21 22 $27625000 $636 23 24 $19152500 $10368 25 26 $1211 $4044 60. 15 43217 8 46490-3 11 2264702-2 GO. 6 104177-2 9 15840087 12 2343381-2 60. 7 12828 10 9486312 13 390946494 316 ANSWERS. P. EX. ANS. EX. ANS. I X. ANS. 60. 60. 60. 14 15 16 47516365-2 7544181-5 6286358-2 17 18 19 749099 1277242 27478 89-3 5 Q9 2 - JO 13957027-5 ft A 1 61. 61. 61. 61. 21 22 23 24 $1126 48 288 13178 25 180607 26 88 27 87066-1 28 $327 29 30 23040 345477 65. 63. 22 23 55 40 24 26 36 27 34 28 54 94 29 30 $8 61 41 64. 64. 32 33 4 and 6 6 34 35 5 II 36 9 II 37 8 38 3 39 9 12 40 7 67. 67. 67. 67. 67. 67. 67. 67. 67. 67. 67. 67. 67. 67. 4 5 6 7 8 9 10 11 12 13 14 15 16 17 194877-24 3283 17359-1 1345 332627-12 795073-41 194877-48 3283 11572-110 2017-108 40367 6704984 78795 10110-9 18 19 20 21 22 23 24 25 26 27 28 29 30 31 3097-33788 307140-121 34960078-346 80496-11707 1672940-165534 206008604-24 30001000-6347 9948157977-81605 1935468-14976 15395919-12214 14243757748-35411 15395919-12714 3008292243-50442 123456789 68. 68. 2 3 132 4871000 4 718328 5 7128368 6 918546 69. 69. 69. 69. 1 2 3" 4 3175 106725 2187600 17624075 1 2 3 4 3550 4700 59250 880300 1 2 3 4 29654200 24678733-1 177925200 74036200 70. 70. 70. 70. 70. 1 2 3 4 1 3704000 1099588000 121300750 88036750 127 2 3 4 5 6 4269 7 87504 8 97049 9 70496-20 10 326 11 284 4741 70424 675 19626 12 IS 14 1 U 237756 1210811 750 24063 54069 71. 71. 1 2 105 387 3 1 133 5 4 1 201 6 387 1935 7 8 1809 12864 ANSWERS. 317 p. EX. ANS. EX. ANS. EX. AN8. 72. 2 17085-29 5 3095-87 8 245-14 72. 3 67639-21 6 45561-16 9 405-141 72. 4 6129-11 7 1392-27 73. 1 4976-3 3 496-321 73. 2 76412 4 6-4978 74. 74. 74. 2 3 4 4-146327 146 91-135803 5 6 1 156557-34400 253-21700 2247-26649 1 900 75. 75. 75. 75. 75. 2 3 4 5 6 329 $45 85 84032 $312 7 g 1C 11 loses $26 $23 276 U4-H $17376 12 13 14 15 16 75 $55 351 85-148 15 76. 76. 76. 76. 17 18 19 20 552 21 lost $1625 22 9 23 2313 24 4285 4562 281 $514 25 26 27 28 6178-6494 $42 $3408 794-f 77. 77. 77. 77. 29 30 31 32 20 36 $2 g. UOcts 33 34 35 36 157-185 12923-13763 $3 5 37 38 $140 $60 78. 78. 78. 78. 39 40 41 42 3750 146 886144 22886826 43 44 45 46 6000-9000 gained $12 Hi $456 t 47 48 6480 8800 82.1 82.' 82: 82. 82. 82. 82. 82. 3 4 5 6 7 8 9 10 37378 375999 670 54000 12500 40000 ; 400000 37500 ; 37 looo 11 12 1 2 3 4 5 6 40368 71453 $67.897 $104.698 $4096.042 $100.011 $4.006 $109.001 7 8 9 10 11 12 13 $0.652 $0.002 $1.607 170.464 $8674.416 * $94780.90 $74164.21 84. 84. 1 2 $73.436 $219.614 3 4 $132.475 $99.11 5 $52.371 318 ANSWEKS. P. EX. ANS. EX. AN8. EX. To" Aim 85. 85. 6 7 $1843.94 $656.369 8 $22.334 9 $7.952 $14.405 86. 86. 86. 86. 86. 6 7 8 9 ( $5.999. $9.742. {$0.744. $87.345. $106.524 $170.056 $44.377 1 2 3 4 5 $2812.50 $51.997 $2.50 $945.361 $2906.961 6 7 8 $24.625 $12.43 $59.827 87. 87. 9 10 $343.675 11 $112.442 12 $8279.155 $932:802 88. 88. 88. 88. 3 4 5 6 $20.35 $375 ' $116.875 $22.95 7 $79.75 8 $3835.625 9 $975 11 $1157 12 13 14 $82.25 $11.25 $510.295 1 89. 89. 89. 1 2 3 $172.50 $168.75 $28.00 4 o 7 $8.40 $45.333 + $357.75 8 9 $3718.50 $104 90. 90. 1 2 $21.40 $60.142 + 3 4 $6 .117 + $8.40 5 $105.026 91. 91. 1 2 $5.961 + $23.597 3 ($3.99. $5.04. $6.6528. $4.3512. $7.8750. 91. 91. 3 4 $3.51 $3.842 + 5 6 $0.06 $0.666 + 8 $16.803 + $41.904 + 92. 92. 92. 92. 92. 9 10 11 12 13 $0.65 $2.12 $0.375 $1.125 $0.14 14 15 10 "^7 18 $3.50 $23.076 + $450 $75.385 $25.25 19 20 21 22 23 $66.666 + $2.50 $24 $0.60 $3.00 93. 93. 93. 93. 24 25 26 27 15tons 25| 9 20 28 32i 29 16 30 112 31 12 32 33 34 Qfi ifi 1 40 93. 93. 93. 1 2 3 $28 $130.50 $51 4 5 6 $12. $0 $0.625 50. $100. $625. .87J. $5.25. $7. Ill $1 ,14 75. $14. $210. $80.50. %[| 9 | $0.06 10 |$14.50. $1015.00. II 11 30 ANSWERS. 319 P. || E, :. ANS. ||EX. 1 ANS. ||Ex.i ANS. 94. 94. 94. li 1 I- 2 4 yds. 6 yds. 3 $414.75 i 3480-$4.50 15 16 j 17 $2.331 18 U1000-5500 19 $23.16 20 155/6s. $547.92 $916 95. 21 |$27.685 || 22 [ $290.82 |j 23 ) $90277.70 99. 99. 2 3 30183/ar. 84226/ar. 4 5 391679/ar. 84 6 1 7 25 12s. 3 d 14s. Id 100. 100. 1 2 60m. 120in. 192m 12yd 18yd 32yd 3 1 4 2 2ft. Kft. 4ft. 4fur. 48/M7-. 64/wr. 101. 101. 101. 101. 101. 33 43 53 68 7 2 16767/*. 59ml 7/ur. 28rd 796602/*. 201 miles. 40700858m. j 109 2Umi Wur. Ird ( %\yd. 2ft. Sin. 1 4na. 16na. 32na. 96^. 128na. 3 30r. 40r. 50r. 9gr. 10<?r. 10*2. 102. 102. 3 { 4 ( 5$ >23na. J04yd 3?r. 2na. 7 95^7. E. qr. HE 103. 1 ( 288m. 432m. ( 864m. 1152m. 240P. 120P. 640.R. 3 160P. 320P. 9yd 1280. 104. 104. 104. 104. 104. 104. 1 2 3 4 3 4P. 16P. 20P. 80en, 160cA. 240cft. 16P. 64P, 96P. ( 20sg. ch. Wsq. ch. \ lOOsg-. ch. 120s^. ch. 3157P. 4 762300. 5 : 260s0./ 16s0. in. 6 93J. 2P. 12P. 7 35if. 563-4. IE. 19P. 8 $12584,25. 9$15,25. 105. 105. 105. 105. 105. 105. 1 2 3 4 1728 Cu. in. 3456 Cu. in. ' 5184 Cu. in. ' Ncu.ft. 54cu.ft. WScu.ft. U2cu.fi. 2G.ft. C.ft. 4SC.ft. 256cu.//. Q4cu. ft. 32cw. #. 5 2cu. yd. 3cu. yd. 6 3 T. H n rn A rn ssa i2a lea. 100. 10G. 100 3 4 5 592704. 200 C. ft. 3200ctt./fc. 5 cords. 2 cord ft. 6 21870 cords-4:C.ft. t. (88 tons. 24ew.y2. ( 1228cw. in. 107. 107. 107. 107. 1 ] 2* 3< 41 (Qgi. Qpt. 12pt. ISpt. 20/rf. S \qt. Uqt. Z4qt. I6pt. 40p^. 4 12^.' 20^.' 80^. 1260*. 2520*. 6 12602^. 10 tuns 2hhd. 25 tuns Igal. $36.64, 320 ANSWEES. p. | EX. ANS. || EX -| ANS. 108. 108. 108. 1 2 3 &pt. lOpt. I2qt. l&qt. 36 qt. 13672^. 4 5 6 12734;^. IWhhd. IZgal 4 5 liar. *lgal. 109. 109. 109. 1 2 I6qt. 40^. Mqt. Zpk. 4pk. Spk. ' 4 ?! 46u. 86w. 106w. Ibu. 1086ii. 1446w. 110. 110. 110. 110. 3 4 5 6 23808p S44pk. 2726w. Ich. 296i*. Zpk. Qqt. 1 2 3 4 64o2.' Icwt. 2qr. 5 3 tons. 111. 111. 111. 111. 111. 3 2790366 drams. 4 90313602. 5 \Udr. 628T. 4cwt. Iqr. 21/6. 7 8 g 10 6r.2c?^.4/6.13oz.l4rfr. 299812802. 212 T. Ucwt. Iqr. 7/6. $118.995-$10. $431.68-$160. 112. 1 48#r. 72(/r. 96#r. 4 1/6. 102. IQpivt. 10(/r. 112. 2 2pwt. %pwt. 5 25/6. 9oz Qpwt. 20^/r. 112. 3 2oz. 3oz. 6 678618^ r. 112. j 48oz. 144oz. 108oz. 7 36/6. 702 112. 4 ( 84oz. 8 38901#r. 112. 5 2/6. 3/6. 8/6. 9 6496ar. 112. 3 148340yr. 10 $657. 113. 113. 1 2 40o/?\ 60^3 \ SOgr. 120(/r. 153 3 40 3 8. 114. 114. 114. 114. 3 4 5 6 80113. 91133. 27ft) 9 94ft) 11? 63 13- 13. 7 8 73918or o (12ft) 8 J23 ,&" 115. 115. 115. 115. 1 2 3 4 72/ir. i20/ir. 1927ir. 168/ir. fiwk. 3 4 5 379467 108sec. 24yr. Ida. 26m. 58sec. 116.11 c \ ( 9?/r. \da, 17/ir 116.11 1 | 16m. 45sec. 7 1 6600/ir. ISOsec. 240sec. 300s?c. ANSWERS. 321 p. EX. ANS. EX. ANS. 116. 2 360wi. 240m. 300i. 3 7 44' 54" 110. 3 120 180 210 240 4 Ic. 5s. 28 15' 116. 4 4 12 3s. 5s. 6s. 5 3946800sec. 116. 1 10765' 6 921625sec. 116. 2 2592000" 7 2 23' 9" 117. 1 57953/ir * 6 Dlb 8 13 23 19prr. 117. 2 10800' 7 1/6. 8oz. bpwt. \%gr. 117. 3 1296000 Cu. in. 8 340157yr. 117. 4 714 9 Sig. 7s. 117. 5 3T. Icwt. 20/6. 10 207^. E. Zqr. 118. 118. 112 12 ( ,320 half pints. >539276dr. 24 C 84mi 3/wr. 4?*d. 1 3yd. yt. 118. 118. 13 | 6-4. IB. 24P. 25 ( 5 A. 35. 35P. 3Jyd. ( 2//. 5in. 118. 141 . pound. 26 1971110251T 118. 15 >7953/*r. 27 26880 times. 118. 16" [s. 15 24' 40" 28 $93024. 118. 17] .2 cords. 29 $27. 118. 18 I244160<7w. in. 30 4 miles. 118. 191 UMHMjpfc, 31 40 yards. 118. 20: 57 7 yd. 2qr. 32 5??io. 3to^'. 5da. 16^r. 118. 21' 1897601. 33 1008 bottles. 118. 22^ 1786024328?. in. 34 110592. 118. 23 L5359/ar. 35 38 casks. 119. 36 17097^ times. 38 248 wu'fes. 119. 37 1013299200sec. 39 $39.879. 120. 2 ; 11377 4s. IJd. 8 203 13 lOgrr. 120. 3 . 1616 7s. 6|d. Q j 79ci6f. 2qr. 18/6. 120. 4 321/6. 802. Ipwt. 190r. } 15oz. 11 dr. 120. 5 J62/6. 602. lOpwrf. 2<?r. in I 340T. 5cto^. 2$r. 120. 6 104ft, 3 3 33 23 40r { 2016. 2oz. 120. 7 55? 73 23 17<?r. 121. 11 16cMrf. 2dr. 1 A ( 1847^. E. 4or. 2na. 121. 1 9 ( 432L. 2ml 4fur. 10 1 1 Jiw. 121. L \ 39rrf. 4yd. j 2639. yd. 5. s^./jf. 121. 1 3 | 1/ur. 34rd. lyd. { 116s<?. in. 121 10 { 1/15. 4iw. 1 8 (27Jf. 27 7 J. 15. 121. 14 - 424 #. Fl. Oqr. 3?ia. lo j OP. 24^. yd. 121. It > 42yd. 3</r. Ina. 19 159^1. 25. 5P. 21 322 ANSWERS. EX. ANS. EX. ANB. 121.1 9ft j 17 Sou. yd. IXC. ft. 21 90cw, 106 C.ft. 121. ay [ 614cii. in. 22 151 a 3a/tf. 122. 23 627/iM. Iqd. \qt. Ipt. 9 j 50w;&. 4c?a. lAr 122. 91 j 50/im Op. Ihhd. SSgal. (41m. 34sec. 122 at ( %qt. j Idr. 9s. 28 33' 122! \ 9^ ( 94c/L 226w. 3pyt. Iqt. ^9 { 59" 122. 122. ( 259cA. 126^. Qpib. Qqt. 30 ( Icir. 10s. 27 2 ; 122. 26 \ Ipt. i j 291/6. 602. 15p7. 122. 199 27 j 172?/r 2mo. Iwk da j. ( 22^/r. i.MM ( ** 123. 2 244/6. 5oz. pwt. 3gr U5A 3A\ 31P 123. q 82 T 16c^. O^r. o \3S%Sq.fl. WSq.in. 123. o 16/6. loz. 7dr. 9 1586tt. Op/fc. 4^. 123. 4 IT. Ocw^ 3^r. 17/6. 10 2T bcwt. 2qr. 21/6. 123. Ooz. 5dr. 11 85yrf. 123. j 336A IH. 31P. 12 3/6. loz. llp^. 17^r. 123. 1 21Q Sq.ft. l36Sq in. 13 322mi Qfur llrd. 123. 6 llQT.llcu.ft.lUcu.in, 14 WQA. IE. 13P. 123. 7 IQSbu. Qpk. 2qt. 3 | 174/6. loz. Ipwt. 3gr. \\ 4 | 8/6. lOoz. 14pu;/. grs. 125. 125. 125. r>^ 5 6 (5T. Icwt. Iqr. 23/6. 1 lloz. j Icwt.Zqr. 20/6.1 loz. I ^i/fr 7" 8 124 T. Qhhd. Mgal. j 14?/r. 46?^-. 4r/. ( 20/i. 58w?. , r )4.sw. 125. 125. 2 3 \)t/r. 4??io. 2rfa. 2lyr. 9mo. 5c?a. 4 17*/>v Imo. oda. 126. 126. 126. 126. 126. 126. 126. 126. 126. 121). 15 7 1 2 3 4 5 6 12yr. 3mo. 26c/a. 22^r. 30yr Imo. 29c?a. 12^r. j 27mo. Zwk. Qda. I 20/ir. 20m. 84?/r. llmo. OM;^. 5c?a. ^2 178. 1/6. 1102;. IQpwt. gr. 6Bb 10 53 13 C7T. IScwt. Iqr. 4/6. { Ooz. %dr. 7 8 9 10 11 12 13 14 2??w 4^/r 2 Ire?. 7yr 9mr> l^a. 362yr. 9 wo. Hrfa. 68/6. lOoz. 3pwt. I5gr. (IT. llcwt.iSqr. \ 7/6. 14oz. 2dr. (84ft 93 43 (13 Ugr. 3?/c/. 2qr Ina. \in. 4 cords 50 Cit6?'c t /<^. ANSWERS. 323 p. EX. ANS. EX. ANS. 127. 15 74 16s. 5d. 2far. 22 16ctd. O^r. 6/6. 2oz. 127. 16 866w. Ipk. Qqt. Ipt. OQ ( 12cw;^ O^r. 23/6. 127. 17 Whhd. 46ya/. Zqt. 29 | 12oz. 127. 18 4366u. Ipk. 6qt. Ipt. 24 14L. Imi. Ifur. I5rd. 127. 19 Icwt. 2qr. 14/6. 25 HA. 3R. 18P. 127. 20 27 Os. lld ( 56z/r 5??io. 27rfa. 127. 21 22/6. 4oz. Qpwt: 13gr. j 3/ir. 25wi. 128. 3 5Qmi. 5>fur. 4rd \\ 4 |27s. 28 22' 45" 129. 5 32?/r. 3mo. 18rfa. 18/ir. 14 122ttu. 4/Ur. 20rtf. 129. J532 7 . Zcwt. 2?r. 16/6. 15 111.4. 2.R. 25 P. 129. { 4oz. Mr. 16 267 yd. O^?'. 3na. 129. 7 256w. Zpk. Iqt. 17 477y. Imi. 7/wr. 8r<f. 129. 8 ZT. 5cwt. Qqr. 24/6. 18 95/i7ic?. Qgdl. 129. 9 8 cords 6 cord ft. 19 32/61 9oz. 15pitl 129. 10 11 yr. 5mo. 3da. 20 746??ii. 5/^ur. 129. 11 3/6. 3oz. \2pwt. 21 15 129. 12 IT. \cwt. 2qr. 12/6. 99 22 ( 56 T. Hcifi. 3^r. 129. 13 13ft) 7 23 I34.gr. ] 15/6. 130. 23 5 4s. M. 29 1493wn. 2/wr. 130. 24 Mhhd. 22gd, Iqt. Ipt. 30 62966w. 3p^. 4qt. 130. 25 927i/cfs. 31 3174 mi/es. 130. 26 748A QR 38P. - 32 72 tows 16cul 20/6. 130. 27 286yr. lb?io. %wk. 33 282^/c/s. 130. 28 56 f. \1cwt. 27?-. 10/6. 131. 3 \5L. 2mi. 6/ur. 36rd || 4 |26w. \pk. 3qt. 132. 5 2cwt. \qr. 18/6. 3^z. | 16 56u. Ipk Q^qt. 132. 6 5yd. 2^?*. 0-^-?ia. 17S >Ib 4 63 13 16(?r. 132. 7 10^1. 3R. SOP. 18 ( \\yal. Iqt. Ipt. 132. 8 21 9s 8rf. 19 J 2A 27?. 25P. 132. 9 llcwt. 3qr. 18f^/6. 20$ t4??ii. 7/wr. 4rrf. 132. 1 10 Ipyt. 2f qt. 21 ] 4/6. Ooz. 8pi^. llgr. 132. 11 1 2s. 4rf. 2/ar. 22 J \gal. Iqt. Ipt. 132 12 2T 7 7?w;'. 23 4 :bu, 3pfc. 2qt. 132. 13 25/6. 3o^ 8rfr. 24 S Icwt. Iqr. 24/6. 132 li 2 34' 16" '25 i Ibu. Iqt. 132 15 49#a/ 2^. lp. 26J .Icwt. <2qr. 11/6. lf| I 27 | 28 7/6, 12oz. 2dr. 15' 29^ 30$ WgcU. Zqt \-fopt. 20mi. 4/?<r 23rd 324 ANSWEES. p. EX. ANS. |] EX. ANtf. 133. 133. 133. 133. 133. 32 33 34 7s. 3d. Ifar. {24 r'ms 5 qrs. 12 sheets. 63 14s. Sd. Ihhd, ISgal. Ipt. 35 IT. Icwt. Iqr. 19/6. qA ( 600 9s. Sd. (1050 16s. lid. 37 53^2-^da 135. 135. 135. 135. 135. 135. 1 2 3 4 5 6 1 9' time 4m. 36sec. 12/ir. 4m. 36sec. P.M llhr. 55m. Zsec. A.M. 41m. 32sec. llhr. 18m. 28sec. A.M. Whr. 59m. 56sec. A.M. 7 4/ir. 56m. 4sec. 8 4/i,. 56m. 4sec. P.M. 9 2/i. 20m. 4sec. P.M. 10 6/1. Om. 8sec. A.M. 11 ll/i. 6m. 4sec. A.M. 137. 137. 137. 137. 137. 137. 137. 137. 1 3x 2 3x 3 3x 4 2x 5 3x 6 2x 7 3x 8 7x 3 5 2x2x2 2x2x2 3x2 2x2x2x2 2x2x2x2 2x2x2 9 10 11 12 13 14 15 16 7x3x3 19x2x2 O 3 and 7 3 and 7 2 and 7 2 and 3 and 7 3,5, 7 138. 2 9 || 3 | 6 || 4 5 || 5 6 || 6 | 5 || 7 14 | 8 | 42 139. 1 24 | 2 | 4 || 3 45 || 4 630 || 5 267 | 6 396 140. 7 12 8 8 || 9 | 4 || 10 3 __ 141. 141. 3 4 840 147 5 6 840 196 7 78 8 84 9 1008 10 156 ] 11 | 223839 1 142. 3 63 4 126 || 5 27 6 | 12 144. 144. 144. 144. 5 6 7 8 1 11 55 9 10 11 12 i 8 27 16 13 705 14 8 15 5} 16 80 6 pounds. 17 18 3f 145. 145. 145. 145. 145. 19 20 21 22 23 266w. 40| pounds. 300 pounds. 500?/<fo. 24 36 25 4C 26 51 27 If 28 5 1 gal. bu. gal. yds. t-sfir'ns. 29 30 31 58 boxes. 4s. 3 cheeses. 151. 151. 1 2 ? 3 4 * w 5 -4 6 J * 11 t* H ANSWERS. 325 ANS. p. ||EX. | i&~\\~rrfan$~$r~ 152.|| a I i> , EX. ANS. EX. ANS. A, Ari~ it, w, . IT, -, V- w, w- 152. 152. 152. 1Y> 1 2 3 4- ,A- yf^r- TT, rVr. _80 4(> JT4_ 1 2 3 4. 1- A- ft- 17 5 6 T f^' 7 8' 7 "5. 153.JI 2 | H, ft- ft, T! , M- II I Hf, -III. m fit, 154.1! 3 M. 1^, if, 155. 155. 155. 155. 155. ! 1 6 7 8 9 10 t- i- H- A- i 11 12 13 14 15 A. 16 2 3 4 156.1 156. 156. 156. 156. 156. 156. 5 apples. SyVV, 24, 219|. 31504||ff, 1345, 794THM- HP- ^3006. 157. 157. 157. 157. 157. f 158.1! 3 159. 159 159. 4 ff . ^. -W-, H- W- Hi, fVV W. 1 HI, W, f if 13 H, 14 15 16 f, . A. 326 ANSWERS. p. EX. ANS. ft, I.2.2. ,--, F?EL __^ ANS. , W-ll 9 rVo, w, m- w. w w- T 6 ^, A, A, if tf , W- EX. 10 11 ANS. ( 3G. 5fl ) y u > y o > 90' j I& it, , & It- 161. 161. 161. 161 162. 162. 162. 163. 163. 163. Ht 25JJ- 52|. 20H- 20 1-42628 10 11 12 2 T o- Hi tt 24 96H- 25 H-'if 13! f,H, 161 164 11 . 12 63 A- 14 15 - '- 165. 165. 165. 1 2 3 15~* 3 1-. 5 35. C 421 123. 16f. 7 8 9 8-J-cords. $10. 13^6u. 10 11 12 14ni. 13 $lr>l 14 5-1 lar. 15 |16|. 166! 1 ia * 21. 38|. 3 75. 4 26|. 5 6 $5J. $43|. 7 8 $180. || 9 36|i. || 10 $161 $440. 167. 167. 167. 1 2 3 fV TrV 5 6 f $J- 7 8 1 Ift 2 3 4 fr 81.' 5 6 675. 20f 168- 168- 168. 8 S 10 114. 2 1344. 11 201. 12 51. 13 $90. I/ If 1( I $25. ) 2 ^ons. <fe 7 ' V?7' 19|'$1295. 20 21 \ 5|?/^s. ^60f 169- 16!). KV). tV- Tfy|| T fV 8 mi- 1 1 ^ 071. |10 11 12 $f. 170.i 16. II 3 226 T 2 5 . || 4 941. 513i. 11^/jr.s. ANSWEES. 327 p. EX. ANS. EX. ANS. EX ANS. EX. ANS. 171. 171. 171. 171. 171. 1 2 3 4 5 ft H I 6 T * r . 6 8 9 |10 ? r 7 men. $|. 14 baskets. L 1 1 1 1 ifio. 2 7J/6. 3 ichUd'n. 4 5^-jj- times. 5 9J times. 16 17 18 19 27 miles. 9J turkeys. Z-fayds. 172.11 1 172. 2 H 10 3 4 *& 5 6 if- m- 8 s 9 10 A* 174. 174. 174. 174. 174. 174. 1 2 3 4 5 6 1%' J??ll. 9 10 11 12 u^ir^ TrtrTnj */ar- |fna. 13 14 15 16 17 18 1& TTT8 T: 5TI T"^t --gm 19 20 21 2 53i 175. || 3 | 3hhd. Slgal. 2qt. \\ 4 f< 176. 176. 176. 176. 176. 176. 176. 176. 176. 176. 176. 176. 176. 176. 5 6 Hr l 8 9 10 11 12 13 14 15 16 3qr. 2f na. 3wk. Ida. 9/ir. 36m. j Gfur. Srd. 4yd. 2ft. \ Sin. 3cwt. Qqr. 12/6. Soz. (2da. IZhr. 42m. | 5 If sec. Jghhd. T 4 T mi. 17 18 19 20 21 22 23 24 25 26 27 28 29 30 fe ^4- ^4^' 5 1 3^3 03 12gr. 24^a/. Iqr. 21/6. lOoz. 10c?r. * 2gal. 3gi. 2E. 6P. 4?/d 5ft. 127 T V. 177. 177. 177. 177. 51 6" 7 ] 8 J m. 3/wr. 18rd. 'fur. Qyd. 2ft. 9in. .cwt. 2qr. 2/6. 13oz. Ahr. 59m, Msec. c 1C 11 12 IQgal 2H</ \\pwt. 3gr. 4cwt. Iqr. 12/6. 15oz. 5-fadr. 328 ANSWERS. p. E3 I ANS. E3 ANS. 178. 178. 178. 178. 178. 178. 178. 178. 178. 178. 178. u It u u r it i< 2( 21 $ 75f-6w. ( llcwt. lor. 7/6. 13oz. 1 jllfdr. > 90f f mi. , ( 5da. 207ir. 52m. [ 2^r. 19/6. 14oz. T ^dr. i 56?/d. ) lc'/. Igr. 7/6. 7oz. T ^%dr. 6pw. 15gT. 22 23 24 25 26 27 28 29 30 ( Zwk. Ida. 12/ir. 19/n. 2ml 2/ur. 16rd. tV 1 9s. 3d. loz. Spivt. 3grr. f 8cu7/5. 3gr. 5/6. 13oz, 3/6. 5oz. l&pwt. IQgr. (rd. lyd. 2/2. 5^m. 71 53 23 10.gr. 179. 179. 179. 1 2 3 16/35. 5 1/35. 8' 2/35. 6' 10" 3" 11"' 4 5 6 5/5. 8 r 2" 1"' 15/2. 4' 10" 4'" 1/35. 11' 10" 11"' 7 8 1 1/3?. 6' 5" 5"' '(ft. 10' 1" 9'" 181. 181. 181. 181. 181. 2 3 4 5 6 77//. 87/35. I/ 16/35. 6' 8" 366/5. 8' 3" 20. 5 C7./35. 7 8 9 10 27/5. 8' 6" 105/.5'7"6'" 39<7.33(7w./5. f 46yd. 0/35. J3'8" 11 12 13 14 (1C. 40. ft. { 3 cu.ft. 158c. yd. 17c./5.4" ^19,64 ^4/15. 4' 6" 185. 185. 185. 185. 185. 185. 1 2 3 4 5 6 3 016 0017 32 0165 18.03 71 81 99 102 14 21 2.009 6.012 565 2.1 1.3 6.000003 3 4 5 6 7 8 5.09 65.015 80.000003 2.000300 400.092 3000.0021 c 1] ll 47.00021 1500.000003 39.640 .003840 .650 188. 188. 188. 1 2 3 1303.9805 428.67789S 169.371 4 5 6 1.5413 444.0924 1215.7304 7 8 9 246.067 389.989 71.21 101494.521 11$641.249 12.111 189. 189. 189. 189. 190. 190. 190 190. 190. 190. 14 IE If 4.0006 $129.761 $1132.365 $16.3275 17 18 19 20 $1033.6279 $51.451 1.215009 23001044.500059 21 22 23 24 .560596 $7.978 $417.563 74435.0309 1 2 3 4 5 6 3294.9121 249.72501 9.888890 395.9992 999 6377.9 7 8 9 10 11 12 365.007497 20.9943 260.4708953 10.030181 2.0094 34999.965 14 If 1C n 4238.60807 126.831874057 63.879674 106.9993 1.1215 .001 ANSWERS. 329 p. EX ANS. EX ANS. EX ANS. 191. 191. 191. 191. 191. 1 2 3 4 5 .036588 .365491 742.0361960 .001000001 .000000000147 6 7 8 9 10 9308.37 311.2751050254 .25 .0025 .0238416 11 3.04392632 12 $17.2975 13 $14.274 192. 192. 192. 192. 14 15 16 17 $4.543944 .0036 240.1 $56.764 18 $46.95 19 $1.051279 20 .00025015788028 21 2.39015 221 23 24 000016 000274855 00182002625 193. 193. 193. 193. 193. 193. 193. 193. 193. 193. 193. 1 2 3 4 5 6 7 1.11 4.261 33.331 1.0001 4123.5 1175.07 f 12.52534 1 125.2534 -j 1252.534 1 12525.34 [ 125253.4 8 9 10 (16.21987-1621.987 -{16219.87-162198.7 (1621987. (20.81100-208.1100 -52081.100-20811.00 (208110.0-2081100. ( 127.3673874-12736.73874 -J 127367.3874-1273673.874 (12736738.74-127367387.4 194. 194. 194. 194. 2 3 4 5 21940. 30100. 1000. 66.666 + 6 2 (10.-100.-1000.-30. 420.-2000.-12.-120. (1200 .3333 + 3 8.3111 + 4 1.563 + 5 1.160 + 1 195. 195. 195. 195. 195. 195. 1 2 j 4 5 $. $. 1- 1! .0 00638 + 10486 + L941 + 3.119 + 8333 + f .25o/"3.26=.815 . j and .034 o/"3.04 * 1 =.10336 .815-f- [.10336 = 7.885 + 7 .0470204 + 8 188Zm. 9 10 11 12 13 '14 1 8.022 +bu. $0.7909 + 1196.172 + 1000. 227. 31 3001 yds. 125.101 Mb*. 196. 196. 196. 196. 196. 1%. 15 $48.141 16 $10055.3025 17 $934.699 18 $46.875 19 $4070.316 20 $16.63 21 22 23 24 25 26 112.29eit. yd. $45.401 $313.313 $0.75 $122.766 + 177ftar. 27 15.68 + &a?'. 28 92<7G/. 2919.8rfa. 30 ! $54.72 31 1725.15Z6. 197. 197. 1 2 .4285 + .88235 + 3 4 .08571 + .25-.00797 1- 5 6 .025-.7435-.003 .5-.0028 + 330 ANSWERS. p. |M ANS. II EX. ANS., || E x.| AN8. 197. 197. 197. 197. 7 1.496 + 8 1.333+.162 + .792 9 .85 10 ,075 ' 11 12 13 14 136 00875 2976 00687c 15 16 17 > 18 .01171875 .135546875 .0001 .222464 198. 198. 198. 198. 198. ToTT* 23067 TTTUo* 3 W- 4 TTrooTTIr 1 2 3 4 5 .02734375/6 108333 + .00035^. 1.3125p&. 6 7 8 9 10 ife 199. 199. 199 199. 199. 199. 199. 199. 199. 1 2 3 4 5 6 7 8 9 12.00384(7?-. 2<?r. 12/6. 8oz. %qt. \pt. 6s. 9d. &cwt. 3or. 8P. Ihhd. 4^ gal. \qt. 6gal. Zqt. 136do. 2 Mr. 10 11 12 13 14 15 16 17 Is. 8^7. 3$r. 1] 19/ir. i loz. 8c 1 Os. 1 17. 1* 1/6. 2ln Ir. 237 11 ?. 7 far. i. 36sec. . *[fl. 11. ir. 59m. 12 48s<?c. 200. 200. 200. 200. 200. 1 2 3 4 5 4.889955M>&.+ 2.4694/6.+ 1.25yd, 1.046875/6. 5.0833.L.+ 6 i 8 9 10 4.8906256M .472916/6.H .78875T. 5.88125^. .0055T . ] h 1 ] 1 .42859226^. 2 .39201c7z. 3 7.8781253/: 203. 203. 6 7 $36.428 $21.25 8 $28.333 + 9 $32.812 + 10$30.833 11|$62. 1 12 13 $3.111 + 24 pounds. 204.j|15|472,50||16|6 days' iork.\\ 17;31i|6M,|jl8 | $18,541 + 205. ! 20 $8. || 21 | $25.50 23 49 men. || 24 | Uwk. 20G.|| 26 | 18 bales. || 27 | \\\ft. long. || 29 1 2*< iays. 207. 207. 3 3 n |3 i 54dcz. 2 10 " 3 4 ( A's g j n $58.33^ 35 1st. 3d. $240 2d. $200 $140 208. 209. 58 1 9|30do. days. - ||40 9rfa.||41|$96||43|72 u-o'72.|j44 42 Georgia. 210. 210. 210. 210 210. 11 21 q 4 5 18 sheep. $112 48da. $6.5625 6 7 8 9 10 126ar. $1.60 $17.273-f 11 - 12 13 14 (3U//.= ( lOJyaT. iOOda. $10 lOmo. 15 16 17 1st. $2.50 2(7. $3.75 3d. $8.75 22i<?a/: 18J/6. ANSWERS* 331 p. sir. 211. 211. 211. 211. EX ANS. EX ANS. EX ANS. EX ANS. 18 19 20 $9.16 ( 1st. 105/6. 12d. 140/6. (3d. 168/6. |21 i 22 23^ 241 ($175 ]<$! ^10500 25 25 26 27 28 29 $18 $83.33| 3 men. 30 36 men [All QI - 515 ' 1 (724 [ hogs. 212.| 212. 212 212'. 32 33 34 ( As $126. B's j $117. C's$72. 546ar. 35 1 36] 37$ 38 S 50.803 + ^1344 55040 39 40 41 1st. $39.20 2d. $19.60 3d. 117.60 $533.331 3 pieces. 216.|| 1 36 || 2 60 II 3 12 || 4 2J II 5 | 24 217. 217. 21 4- v=i- *j oo 2 TOO " 3 " 1 ?XS 4> 8 9 Si! ^ tOj-a li. LI T IJV- T I T 219. 219. 219. 1 2 3 308mi. 4 3300 pounds. $165 5 $61.425 $1381.25 6 10955mi. 7 S 10 9243,7 5 *20 1,75 36000 rations. 220. 220. 220. 220. 220 11 13 14 15 16 Yr. 20m. L861 + 227 12s. ld.+ $115.50 $29.25 117 18 19 20 Sl^ft. 2 140.32 2 (As $1787.50 2 ] 's $1283.75 - 122.50 1 $1.871 2 $0.154 + 3 $6206.931 221. 221. 221. 221. 24 25 26 27 $61.425 5s. 9d. 3156w. 28: 29 30 31 $252 24yd 72 hats 376ar. 32 33 34 35 21f/6. $12.13 28/ir. %% acres 36$] 37$] 38 $j 39 $5 .8.27 68.742 + 08.25 53.125 223.11 1 | 8 days. 224. 224. 224. 2 3 4 27da. 5 72da. 6 160da. 7 20/irses 18da. 8 S 1C 27da. 11 12 13 256 lOd i. 14 1-fclb. a., 22(). 1 | $45 || S 5 150/6.H 3 $99|| 4 232da. || 5 511i??ii'. 227. 227. 227. 227. 6 7 8 9 18yr. 27 weav 72 men. 10 4J T da. 11 11126a. 's 12 2^- tons 13 343 1 ft. 14 15 16 17 15/6. 38fraz's 1926ar. 200 more Myd. long. 229. 229. 229. 229 1 2 3 5 $857.142f A's. $142.857| 7?'s. $480 As. $750 B's. $675 C's. $1500 Mr. Ws. $2100 Mr. J's. $400 ^'s. $800 B's. $1200 C's. 4 ($1866.66| As. -] $1066.66| B's. ($1066.66| C's. 332 ANSWEKS. P. m 230. 230. 230. 230. 230, 230. 230, 230 ANS. $77 A's. $260 B's. $54. As. $38.50 B's. j$60.777+^'.9. $127.633+J5's. 2 | $328.201 + Z>'.s. $1666.66| As. $3888.88f B's. $9444.44f C's. Rs. = $273.365 rcearfy. ^'s= $476.635. (Fuller's $1808.8669+, Brown's $1596.0591 + 1 Dexter' s $1995.0738+, The remainders added ( will give the exact proof. 232. 1 $16.25 7 $8.93 is* ^2109.0392 19 $42.60 232. 2 19.50.yd. 8 18. 06 step 141 575 20 4326ar. 232. 3 39.375cto. 9 $18.5487 15! 5229.08 21 42/zM. 232. 4 $2.375 10 280 cows 161 ;350 22 $24.25 232 5 I55.48mi. 11 892.5 tons 171 ^375 232. 6 5 oxen. 12 1015/6. 18j 694.232 233. 23 $10.80 1 .25 5 .88A 9 16* 233 91 f 26f per ct. left 2 .50 6 .05 - 233. \ 3333.33$. 3 .40 7 :01A *>- 25 $1304 75 4 j 20 Q 0^ 235. || 2 | $24862.50 || 3 $233.75 | 4 | $8443.75)15 | $14700 236. II 9 | 200 shares. \\ 237. 1C ) 80 shares. 238. 238. l 2 I1.06J $0.75 loss. 3 4 $0.966 + $1.00 5 6 $112.50 $208.4375 7 8 12.054 25 per ct. 239. 239. 239. 9 10 18 per c ($13 w } 90 t. hole g'n oer ct. 11 12 13 $1.025 I1.03H. $2.216|. 14 15 ISyd. $9.21 T V ( ^ U i 240. 240. 240. oin 1 2 3 $43.77 $1312.50 j $237.60 ) *1ft8 4.0 4 5 6 $210 $607.50 $1381.80 1 *.^04. 8 $450 9 $1320 $142.95 11 12 13 U $1800 $45 $47.624 + $9558,437 + *fiftno 242. 2 $39 .;; 1427.50 10 $183,9705 2!$121.325 242. 3 $266 7 $9.5067 11 $4454.857 3 1315.389 242. 4 $4446.75 8 $331.1511 12 $30455.0224 4 221.075 242 5 $642.60 9 $1158.0668 l'$95.229 + 5 1290.798 243.JI 2 | $10.8012 j| 3 | $2.728+ ANSWERS. 333 p. ||EX. ANS. HEX. | ANS. || EX ANS. 244 244. 2 3 $309.5034 $35.1485 + 4 5 $30.5598 $14.0979 6 $64.5792 245. 245. 245. 245. 245. 245. 7 8 9 10 11 12 $76.2433 $194.6177 $328.32 $1004.6976 $1183.6935 $1445.2333 13 14 15 16 17 181 $190.148 3286.40 6322.8825 7500.60 75.04 218.88 19 20 21 22 23 24 $600.445 $44.2893 $167.001 $3126.203 $9051.668 $4968.9975 246. 246. 246. 246. 246. 246. 247." 247. 25 26 27 28 29 30 $141.8136 $272.80 $39.9274 $928.0686 $529.925 $31.2681 31 32 33 34 35 36 $94.269 $245.4896 $76.966 $33.3232 $28761.776 $5678.071 37 38 39 40 41 42 $217.5116 $6214.14 $856.690 $383.3808 $188.0349 $3720.465 2 3 15 2s. 8Jrf. 4 24 18s. 3Jd.+ 5 26 10s. 11 331 Is. Qa d i 249. 2 $860.4194 || 3 | $167.983 + 250.||1|$950||2|7 per ^. 251. 251. 2 3 $19.101 $36.50 + 4 5 $404.0625 $291.60 6 7 $211.456 $185.775 252 253 254 254 254 254 254 255 255 256 257 258 ||8|$171.6Q75||9i$118.528||lQ|$315.2438||lli$152.408 | $1750 present value.\\'2 \ $1565.402+ pres. vol. 254. 254. 254. 254. 254. 3 4 5 6 7 $9677.50+ pres. val. 223 5s. 8d. discount. $5620.176 +pres. val. $702.485 $1.94 difference. 8 9 10 11 12 $3869.407+ pres. vol. $2109.236+ " " $2763.694+ " " $4000 " $6.473+ loss. 255. 255. 1 2 $6.3291 $10.50 3 4 $15240.54 $5.8408 5 6 $3393.504 $29.0096 7 $122.81 + 256. 8 | $341.709 + ||1 $344.66 + ||2 $5734.32 + | $695.64||4|$118.85 + |[5|$1740.60||6|376.46 + |2|12mo.l!3|87riQ. day of March. or ANSWERS. p. |EX.| ANS. EX ANS. 2bo. 1 $426.416 (21 5s.-25 14s. 3d 30 265. 2 1073 18s. l\d. 6 \ 17s. IcM 1 2s. 9id + 265. 3 $1967.892 + (38 11s. 4K-2319s.lUrf. 265. 4 389 6s. 2fd. K j $250-$250-$250-$250. 265. 5 $2551.733 \ $516.66^ 4250. 266. 1 $3720.937 3 $6748.60 5 $3643.875 266. 2 $8668.935 4 $4583.94 + - 268. 270T 2 | $1270.428 || 3 || $2016.11 || 4 | $16975.775 |2812.50||2 3 | 1351.45+ || 4 271.111 I 3s.||2 | | .288+c/s.||4 | 73' 274.! 274J 274. 274. 274 274. 3 5 6 1 2 3 1 3 3 4 1 9 4, 8, 2. || 4 | 1/6. 1/6. 3/6. of 16. 2 o/ 18. 3 of 23. 5 o/ 24 jal at 10s.-3 a/ 14s.-4 at 21s. 4 a/ 24s. gal at 4s., 4#a/. at 5s., 8 a/ 5s. Qd., ai 46u. TF. 286it. E. 146it. I?. 286ii. 0. 66w. W 1 2ft?/. 72. 1 2ft?/. /?. 1 2ft?/, O id 8 at 6s. 275 275' 275 275. 4 1 1 2 3 40(/a/. F. 80^a/. E. 20gal. spirits. 10 of Is/. 10 of 2d. 30 of M. 36/6. at d. 36 at 6d. 36 at Wd. 36 at I2d. 21 1 of each. 4 eac/i of the 1st. three and 30 of 15 cora/s je. 276. 276. 276. 276. 276. 276. 276. 276. 276. 276. 1 2 3 4 5 6 7 8 I =1 J - 5 - : hr. *=& 9 =81 12 3 =1728 125 3 = 1953125 16 3 = 4096 9 10 11 12 13 14 15 16 17 18 9 4 =6561 16 5 =1048576 20 6 = 64000000 225 2 =50625 2167 2 =4695889 321 3 =33076161 215 4 =2136750625 = 610437195439776 9 6 =531441 36()49 2 = 1299530401 282. 282. 282 28'I 1 2 3 4 5 1.732054- 3,31662 + 32.695 + 1506.23 + 2756.22 + 6 7 8 9 .10 6031 4698 57.19 + 69.247 + 2091 + 11 12 IS H l |.05 .01809 .0321 2.104 .[2.91547 + |17 18 19 20 0.71554 0.41408 + ANSWEKS. 335 284. 284. 2 i 25/35. 1-26-4 9 rd - + 3 4 85 97.75mi.+ 5 6 82 partners. Jp. 28- 28- 285 288 288 289 289 289 290. 292. EX. ANS. ANS. EX. ANS. 7 | 62 trees \\ 8 9 | 10 | 4.90. 288. 288. 1 2 73 179 3 4 319 439 5 6 638 364 1 2 ,54 .95'5 3 2.35 4 .909 .707 1.505 289. 289. 289. 1 2 3 i 3f t- 4 5 6 .829 + .822 + .873 + 1 2 3 17 28-4704 16.197/35.+ 4 6 14.58/55.+ 1728 12/3L : 6ft. 290. | 8 268.0832 || 9 2/15. 4iw. || 10 | 2ft. II U 12/2. j. I $1.53 || 2 | $212 || 3 | 40 || 4 a 5mi. II f 2 5 ^ )4") ! i ^ 2 2s. 8d. II 3 i 4 || 4 I 78732 || 5 | $25600 || 6 ' $61.44 297. 1 6560 3 381 5 $196.83-$295.24 297. 2 254 4 204 158. 6 $4800-$9450 2 ( J8. 1 $978 6 3. 8 * (213. 3125 15 $3567 298. 2 516i*. 7 Iti L yds. . 1211. 6875 16288 298. 3 $80.71 8 T 3 T7- 12 If and i-i- 17 $4717 298. 4 5467A?\ 9 4^1 13 9.04 18 137 298. 5 $26.25 10 120 men. 14 4 299. 19 108 T 7 5 P l a nks 21 1879000 299. 20 3800 3C 792 299. 21 SQyds. C Iw fur 3 3ro 299. 22 ( A'sjto. &s $60. | C"s $32. 31 -{ ! r, /./ ( loJ/#. 32i62 years. 2 ( J9. 23 7-^- days. 334 299. 24 3??io. ($2454 1st. 299 o _ ( ^Vo- W 34 -^$3681 2d. 291). 1 $986.66| worth. $4294.50 3d. 299. 26 li 35 408 saves. 299. O H 0. %d. $1 20. 36J23 ifci 299. ""' "j 3rf. $14 3 || 28 |50. 37 2400 300 38 3 o'clock 42 32 T 8 T ??*m. j 52</r. 11 mo. 300. 39 300 me?i 43 $34782.608 'Id. lOJfcr. 300 40 864 44 $3.653 49 5-irf. 300. ' J's $2364 45 ^4 A per ct. 50 lll?/r. 300 . 7?'.s$ll 82 46 $7816091 + 51 22500 6ricfc. 300. " O's$788 47 57 pieces. !52 27.7/1 300. Us $304 ! 53 5 years. 336 ANSWERS. p. H ANS. 301. 301. 301. 301. 501. 301. 301. 301. 301. 301. 2250 men. ( $196.83 last terms. j $295.24 whole ain't ( 96fru. wheat. 12 rye. 1 12 barley. 12 oats. 356.25 $8640 $1.20 lost 4 pence. EX | 62 63 64 ANS. 4 days 240 hour*. A 21-Ji SC days. G Y 's $ Z>'*=$77.92}f $1020,66 $8925.544 + 302 302 302 302 302 302 50/. 9mi. 5/ur. 34rd. 9/ j daughter $780. so?i ($3120. wi/e$1560. 76mi.-1292mi. 4?/r. llmo. $423.36 $920.20 1st. $2760.60 74 75 j | 2d. 5521.20 3d. 76 3Ar. 20m. 77j69fm. f'n 303. 303. 304, 304 304 304 400s#. yd. 305. 305. 305. 305. 305. 305. 5 10A 6 7A 2 QA. OR. 12P. 3 | 2 A 3P. 15P. 1 ! 109A IJR. 28P. 1 2 3 4 5 6 12.5664 292.1688 62.8320 25. 3709 2180.41 + 32520^. yd. 45849.485 2 5A1P.9.95P. . 5'7"6"' 28^2744 78.5400 38.4846 1069 + 452.3904 1539384 72^1. 371 24P 3 A 3A 25P. 176.031258(?. yd. 15^. OR. 33.2P. 67 A 27?. 16P. 1809.5616 904.7808 33.5104 1436.7584 2120.58 911064 3392.928 1242 6 3600 2 2290.2264 5 706.86 309.1! 1 i 309 2 196.52^. 3 I 1360of. 115 _185.06885raZ. 4 | 182.844r/a/.|l- s YB 17361 M306011 QA THE UNIVERSITY OF CALIFORNIA LIBRARY 'ERIE SPELLER & WM'SGEL l<To 1 9 Rational Primer, No 2,- -National first Reader., -National Second Header, . 16mo. 4. National Third Meat itional Fourt/ Header 9 , 6* National Fifth Reader, 'onal Elementary Speller, . tonal Pronouncing Speller, . . 482 pp., 1 600 pp., 12mo. 160 pp., 16mo. 188 pp., 12mo. ^EPENDEVT READERS. , EBS. 'ers surpa tionii \liefoli ire: 3UiLDING. FES, ,'til and exclusive. Dtfv . iUNCIATIONc % and exactly treated, nA-Phone ARTICULA' > AND ORTHOEP CATIONS, tipeca, ->iven. PUNCTUATIC , ON , TauffM by I . ... / Lesson* ffo gape* ELOCuTior.', rs, fflafvrate Treatises, More than any ottw ^1* of SELECTIONS. Diversified, Jndidouti cwwt A: '*?*iw ARRANGEMENT, Toy/ nbrate attention* >r the amount t// m LITERATI; -OP The war I a' ' 6 <* . st authors. ->-a4 and Vnj, .Parker &. Watson's Keade, adoj^ted for excl more than